Proceedins of the Symposium on Geometric Function Theory
Organisors A Gangadharan, V Ravichandran
Editors V Ravichandran, N Marikkannan
Sri Venkateswara College of Engineering Pennalur, Sriperumbudur 602 105, Tamil Nadu December 2, 2002
Patrons Prof A Krishnan Mr V Thulasi Doss Prof S Muthukkaruppan Advisory Committee Prof Prof Prof Prof Prof Prof
P Balasubrahmanyam R Balasubramanian G Lakshma Reddy Rajalakshmi Rajagopalan T N Shanmugam K G Subramanian
About the College
Sri Venkateswara College of Engineering is a unit of Sri Venkateswara Educational and Health Trust. The College runs undergraduate and postgraduate programs in Engineering and Computer Applications. The College is affiliated to the Anna University and is approved by the AICTE. The College is ISO certified and accredited by National Board of Accreditation. The College is situated in a quiet environment about 37 Km from Chennai on the Chennai-Bangalore National Highway (NH4), at Pennalur, Sriperumbudur.
Acknowledgement
We thank our Secretary, Treasurer and Principal for encouraging and sponsoring this symposium. Several people helped in the preparation of this proceedings. Notable among them are S Raju, T Kamaraj, R Raghu. In fact, one of the article was typeset by Raju. Credits goes to K Dhanasekaran for the excellent Xeroxing. Thanks are due for everyone who have given permissions to use their facilities.
Dedicated to the memory of Prof K S Padmanabhan
Symposium on Geometric Function Theory Programme
Inauguration 9:30–10:30 • Prayer • Welcome address by Prof. A Gangadharan • About KSP by Prof. M Jayamala • Inaugural address by Prof. R Balasubramanian, Director, Institute of Mathematical Sciences Invited Talks Chair: Prof. M S Kasi 10:45—12:30 • Specialisations and generalisations inside the unit disc by T N Shanmugam 10:45–11:30 • A Differential Operator and its Applications to a General and Novel Class of Meromorphically Univalent Functions with Negative or Positive Coefficients by Prof S B Joshi 11:30—12:00 • Generalisation of Pick-Julia Theorems For Operators In A Hilbert Space by Rajalakshmi Rajagopal 12:00—12:30
Paper Presentation – I Chair: Prof. T N Shanmugam 1:30—2:30 • Some families of Analytic Functions with Negative Coefficients by P Sahoo 1:30-1:45 • An Application of Fractional Calculus Operators to a Subclass of pvalent Functions with Negative Coefficients by Sayali S. Joshi 1:45-2:00 • Contractions of Subclasses of Harmonic Univalent Functions by K Vijaya 2:00-2:15 • On Uniformly Spirallike functions and Uniformly Convex Spirallike functions by C Selvaraj 2:15-2:30 Paper Presentation – II Chair: Prof. Rajalaksmi Rajagopal 2:45—4:00 • A note on a subclass of close-to-convex functions by N Magesh
2:50–3:00
• Meromorphic Starlike functions with positive coefficients associated with Parabolic regions by S Sivaprasad Kumar 3:00-3:10 • On the weighted mean of univalent convex functions of order ρ by Helen C David 3:10-3:20 • Functions with negative coefficients defined by convolution by N Marikkannan 3:20-3:30 • Certain Subclasses of Meromorphic Functions using Convolution and Differential Subordination by A Gangadharan 3:30-3:40 • Vote of Thanks by Prof. A Gangadharan
Proceedings of the Symposium on
Geometric Function Theory (December 2002) Contents
• Specialisations and generalisations inside the unit disc by T N Shanmugam
1–8
• Generalisation of Pick-Julia Theorems For Operators In A Hilbert Space by Rajalakshmi Rajagopal 9–19 • Contractions of Subclasses of Harmonic Univalent Functions by G Murugusundaramoorthy, K Vijaya, Thomas Rosy 21–31 • Meromorphic Starlike functions with positive coefficients associated with Parabolic regions by V Ravichandran, S Sivaprasad Kumar, R Usha 33–40 • Certain Subclasses of Meromorphic Functions using Convolution and Differential Subordination by A Gangadharan 41–48 • Functions with negative coefficients defined by convolution by N Marikkannan, V Ravichandran • On certain functions defined by Ruscheweyh derivatives by V Ravichandran, C Selvaraj, Rajalaksmi Rajagopal • A note on a subclass of close-to-convex functions by N Magesh, C Selvaraj, Rajalaksmi Rajagopal
49–56 57–62 63–68
• On the weighted mean of univalent convex functions of order ρ by T V Sudharsan, Helen C David, K G Subramanian 69–74 • An Application of Fractional Calculus Operators to a Subclass of pvalent Functions with Negative Coefficients by S R Kulkarni, Sayali S. Joshi 75–84 • A Differential Operator and its Applications to a General and Novel Class of Meromorphically Univalent Functions with Negative or Positive Coefficients by Prof S B Joshi 85–95
Proc. Symp. Geometric Function Theory, Dec 2002, pp. 1–8
SPECIALISATIONS AND GENERALISATIONS INSIDE THE UNIT DISC T N SHANMUGAM Abstract. In this paper, how generalisation and specialisations of various classes of functions defined in the unit disc U = {z : |z| < 1} came into being are discussed. Also certain types of problems and inequalities for the classes of Normalised analytic functions in the open unit disc are discussed.
1. Introduction Let A denote the set of all functions defined in the unit disk U = {z : |z| < 1} which has the Taylor series of the form (1)
f (z) = z +
∞ X
an z n
n=2
Let S denote a subclass A of functions which are univalent in the unit disk U . We describe various classes of functions defined by the geometric conditions and give their analytic conditions in the following: Definition 1 (Geometric Definition of Starlike function). A domain is starlike with respect to origin if every line segment from origin to any point in the domain is entirely inside the domain. A function f ∈ S is said to be starlike (more precisely starlike with respect to the origin) if the image of the unit disk is a starlike domain. An immediate generalisation, in view of the definition, of this class is the class of functions which are starlike with respect to any point instead of the origin. Definition 2 (Analytic Condition for Starlikeness). A function f ∈ S is said to be Starlike if it satisfies the condition (2)
Re
zf 0 (z) > 0 z ∈ U. f (z) 1
2
T N SHANMUGAM
This class of functions can be generalised to the class of starlike functions of order α as in the Definition 3. A function f ∈ S is said to be Starlike of order α if it satisfies the condition zf 0 (z) (3) Re > α 0 α < 1, z ∈ U. f (z) Definition 4 (Padmanabhan Starlike function of order α [8]). A function of the form (1) is starlike of order α if it satisfies zf 0 (z) − 1 f (z) (4) < α 0 < α ≤ 1. zf 0 (z) + 1 f (z) Definition 5 (Geometric Definition of Convex Function). A region is said to be convex if the line joining any two points lie in the region. A function f ∈ S is said to be a convex function if it maps the unit disc on to a convex region. Definition 6 (Analytic Definition of Convex function). A function f ∈ S is said to be a convex function if it satisfies the condition zf 00 (z) (5) Re 1 + 0 > 0 z ∈ U. f (z) An immediate generalisation of this class is th class of convex functions of order α. A function f ∈ S is said to be a convex function of order α if it satisfies the condition zf 00 (z) (6) Re 1 + 0 > α z ∈ U, 0 ≤ α < 1. f (z) Note that every convex function is starlike with respect to every point in the domain f (U ), hence every convex function is a starlike function but the converse need not be true. The best example of a starlike function which is not a convex function is the famous Koebe z function k(z) = (1−z) 2. Another important class of functions is the class of functions with positive real part P . A function p(z) = 1 + p1 z + p2 z 2 + . . . ∈ P if 0 (z) ∈ P. Rep(z) > 0. Note that f ∈ A is starlike if and only if zff (z) Next we have the class of functions called Close-to-Convex function: Definition 7. A function f is called a close to convex function if it satisfies Z θ2 zf 00 (z) (7) Re 1 + 0 dθ > −π. f (z) θ1
SPECIALISATIONS AND GENERALISATIONS INSIDE THE UNIT DISC
3
Another way of defining the close -to -convex function is that f in S is said to be close -to -convex if there exists a convex function g such that f 0 (z) (8) Re 0 > 0 z ∈ U. g (z) Note that if g is taken as a starlike function and instead of the derivative of the function only the function is taken in the above we get a close -to -star function. Also by Specializing on the functions g(z), one can get different functions which are close-to-convex or closeto-star. Also observe that there is NO Characterisation of these classes of functions through their images like the one we had it for starlike functions or convex functions. The class of close-to-convex functions include the class of starlike functions and hence the class of convex functions as a sub class. A larger class of univalent function than this class was introduced by I.E.bazilevic[1] . Definition 8 (bazilevic[1). functions] A bazilevic[1] function of type (α, beta) is of the form Z z 1 tiβ−1 g α (t)h(t)dt} α+iβ (9) f (z) = {(α + iβ) 0
where g(z) is a starlike function, h(z) is a function with positive real part, α > 0, β is any real number. It was I.E. bazilevic[1] who proved every such function is single valued and univalent in the unit disk U. Analytically a function f in A is a bazilevic[1] function of type (α, β) if and only if eiλ f α+iβ−1 (z)f 0 (z) ∈P g α (z)z iβ−1 for some starlike function g and for a real λ.
(10)
The class of bazilevic[1] functions includes the following subclasses: (1) K — the class of convex functions (2) St — the class of Starlike functions (3) C — the class of close -to -convex functions (4) B(1, 0) — the class of bazilevic[1] functions of type (1,0). Definition 9 (subordination). Let f and g be two functions defined on the unit disk U . We say f is subordinate to g if f (0) = g(0) and f (U ) is a subset of g(U ). We write this as f ≺ g. The class P of caratheodory functions can be generalised using subordination. Note that a Riemann mapping of the unit disc onto the
4
T N SHANMUGAM
Right Half plane with p(0) = 1 is 1+z and hence the class P of functions 1−z with positive real part or Caratheodory functions can be expressed as those functions p(z) which are analytic in the unit disc with p(0) = 1 . This way of realising the caratheodory class and satisfying p(z) ≺ 1+z 1−z paved way for a generalisation by W. Janowski[5] as follows: Let −1 ≤ B < A ≤ 1 . Denote by P (A, B) the class of all functions p 1+Az analytic in the open unit disc U with p(0) = 1 and p(z) ≺ 1+Bz z ∈ U. Note that when B = −1 and A = 1 the class P (A, B) is the same as the caratheodory class. For various specialisation on A&B we get several subclases of P . This class was further generlalised by replacing the function f rac1 + z1 − z by any convex univalent function h(z) with h(0) =1 and real positive function as the class P (h) which is the class of functions p analytic in √ U and p ≺ h. Taking a special case as h(z) = 1 + π2 log( 1+1−sqrt(z) ) one z gets the class of functions which are parabolically real part positive as the the above function h(z) is the Riemann mapping which maps the open unit disc onto a parabolic region symetric about the Real axis with the vertex at the point z = 12 0 (z) 00 (z) 0 )0 Cosequently by requiring zff (z) , 1+ zff 0 (z) , (zf in the class P (A, B) g 0 (z) or P (h) we have the corresponding genralisations of the classes of starlike ,convex ,close -to -convex functions etc., As a general refernce to this kind of study the reader is refered to the following two books by P.L. Duran[ 3 ] and A.W.goodman [ 4 ]. 2. Coefficient Problems If f ∈ S has the form (1) , then finding an estimate for |an | is the coefficient problem. It was conjectured by Bieberbach that for f ∈ S, |an | ≤ n. This was proved affirmatively by Louis de Branches in the year 1984. When f is a starlike function the same inequality holds. When f is a convex function then |an| ≤ 1 . Such questions had been answered for several subclasses of S and still research is going on for some subclasses of functions. Bieberbach gave another coefficient estimate for functions in the class S as follows If f ∈ S has the form (1), then |a2 2 − a3 | ≤ 1. Marty relation for coefficients of functions in the class S. Let f be a function in the class S . Then f (z) has the form (1) and the coefficients satisfies the relation (11)
(n + 1)an+1 − 2a2 an − (n − 1)an−1 = 0,
SPECIALISATIONS AND GENERALISATIONS INSIDE THE UNIT DISC
5
provided |an | is the maximum for all functions in the class S. Problem. If f (z) is any analytic function in the unit disc and has the form (1) and satisfies the Marty relation for every n, is the function f univalent and an extremal function? 3. Integral Operator If f is a function inR a class S, An integral transform of f (z) is of z the form F (z) = k(z) 0 g(α, β, t)f (t)dt , where k(z) is a Normalising function which will make the function F (0) = 0 = F 0 (0) − 1. The question is: whenever f is in some favored class is F (z) also is in the same favored class?. This question arose after M.S Robertson (1964) proved that the function Z 2 z t F (z) = dt z o (1 − t)2 is Univalent . Hence this paved a way fo the generalised discussion of the various integral operators as follows Libera[6] (1965) Integral Operator Z 2 z f (t)dt F (z) = z 0 Bernadi[2] (1969) Integral Operator Z α+1 z F (z) = f (t)tα−1 dt α z 0 Note that when α = 1, this operator reduces to the Libera[6] operator. β1 Z α+β z β α−1 F (z) = f (t) t dt zα 0 Rushweyh (1973) Integral Operator Note that when β = 1, this operator reduces to the Bernadi[2] operator. Mocanu[7] (1983) Integral Operator Z z 2 g 0 (t)g(t)dt F (z) = g(z) 0 Note that if g(z) = z the this F (z) is the same as the Libera[6] Operator. shanmugam[10](1987) Integral operators α Z c + α1 z c−1 1 0 g (t)g (t)f α (t)dt Ag (f )(z) = g c (z) 0 + This operator generalises both the Rusheweyh and Mocanu[7] operator.
6
T N SHANMUGAM
4. Convolution P Let f (z) = z + an z n and g(z) = z + an z n be in S, Ptwo functions n then their convolution is given by f ∗ g(z) = z + an bn z . Note that the Integral operators defined by Libera[6] and Bernadi[2] can also be represented interms of convolution of the function g with the function f in this case the function g is given by P
g(z) = z +
∞ X n=2
2 n z n+1
∞ X α+1 n g(z) = z + z α+n n=2
respectively. This gives rise to the following question. If f and g belong to some favored class is the convolution of f and g also belong to that favored class of functions? More particularly when g(z) is a convex function the above question was answered affirmatively by Rusheweyh and Shiel-small[ ]. From then on these types of questions were rised and answered either wholly or partially for several sub classes of S. Problem: If g(z) is a close-to-convex function, is the same result hold good? ¯ 5. Radius Problem The problem is to find the maximal radius r < 1 so that a function f has the property P in Ur = {z : |z| < r} or f (Ur ) has the property P. If f is a starlike function then we know that f need not be convex. The maximal radius r so that the image of the disk Ur under such function is convex is called the radius of the convexity of the class of starlike functions. Simillarly there are problems of finding r so that (1) If f is a function in some class imply that f is in some other class whenever |z| < r < 1 . (2) Finding the radius r so that the integral transform F of f is in the same class , if it is not in the same class . (3) if f and g is in some class of functions then to find and r so that f ∗ g also in the same class whenever |z| < r < 1. 6. Faber polynomial and Grunsky Inequality P The Class consists of all functions of the form g(z) = z + b0 + P∞ −n which are n=1 bn z Panalytic and univalent P in the domain ∆ = {z : |z| > 1}. The class P0 is a subclass of P consisting of all non- zero functions. The class 0 is a subclass of with b0 = 0.
SPECIALISATIONS AND GENERALISATIONS INSIDE THE UNIT DISC
7
P Theorem 1 (Relation between the class S and ). f ∈ S if and only P P P if g(z) = f (11 ) ∈ . ∗ is the subclass of consisting of functions z whose omited set E has a two dimensional lebesgue measure zero. For example the omitted set could be an arc of a curve. P Theorem 2 (The Area Theorem). If g ∈ , then ∞ X
n|bn |2 ≤ 1
n=1
and equality hold if and only if g ∈
P∗
.
This is called the area theorem because it is proved using the area of the image of the disc |z| = r under g is X π{1 − n|bn |2 } ≥ 0. This theorem has a generalisation as a Theorem 3 (Polynomial area Theorem). Let g ∈ non-constant polynomial. Let p(g(z)) =
N X
P
and let p be any
ck z −k
k=−N
for |z| > 1 where N is the degree of the polynomial p(z). Then ∞ X
k|ck |2 ≤ 0.
k=−N
Whenp(w) = w, we get the ordinary Area theorem. P Theorem 4 (Faber Polynomial). If g ∈ , then the expansion ∞
X zg 0 (z) Fn (w)z −n = g(z) − w n=0 is all z in some nbhd of ∞. The functions Fn (w) = wn + Pnvalid forn−k is a monic polynomial called the Faber Polynomial of k=1 ank w the function g. Since g is a univalent function the function ∞
∞
XX ζg 0 (ζ) ζ βnk z −k ζ −n − = g(ζ) − w ζ − z n=1 k=1
8
T N SHANMUGAM
is analytic in the region |z| > 1, |ζ| > 1. Hence putting w = g(z), we get ∞ ∞ X X −n n Fn (g(z))ζ = 1 + {z + βnk z −n }ζ −n . n=1
n=0
Thus the Faber polynomials satisfy n
Fn (g(z)) = z +
∞ X
βnk w−k
k=1
for n = 1, 2, 3, . . .. The coefficients βnk are called the Grunsky co efficients of g. Theorem 5 (Grunsky Inequalities). Let βnk be the Grunsky coeffiP ceients of a function g ∈ . Then 2 N N ∞ X X X βnk λn ≤ n|λn |2 k n=1 n=1 k=1
for every positive integer N and for all complex numbers λ1 , . . . , λn . References [1] I E Bazilevic, On a case of integrability in quadrature of the Lowner-Kufarev equations, Math. Sb. 37(79) (1955), 471–476 (Russian). [2] S D Bernadi, Convex and starlike univalent functions, Trans. Amer. Math. Soc., 135(1969), 429–446. [3] P L Duren, Univalent Functions, Springer-verlag, (1983) [4] A W Goodman, Univalent Functions, Vol.1 & 2, Mariner Publishers, Tampa , Florida. [5] W.Janowski, Extremal problems for a family of functions with positive real part and for some related families, Ann. Polon. Math.23 (1970/1971), 159–177. [6] R J Libera, Some classes of regular functions, Proc. Amer. Math. Soc., 16(1965), 755–758. [7] P T Mocanu, Some integral operators and starlike functions, Int. seminar on functional equations, approximation and convexity (Cluj-Napoca, 1983), 91–94, Preprint, 83-2, Univ. ”Babe¸s-Bolyai”, Cluj-Napoca, 1983. [8] K S padmanabhan, On certain classes of starlike functions in the unit disk, J. Indian Math. Soc. 32(1968), 89–103. [9] St. Ruscheweyh and T Sheil-Small, Hadmard products of Schlicht functions and the Polya-Schoenberg Conjecture, Comment. Math. Helv. 48(1973), 119–135. [10] T N shanmugam, Some studies on analytic functions with special reference to integral operators, PhD Thesis, Anna University, (1987). School of Mathematics, Anna University, Chennai 600 025, E-mail address:
[email protected], WWW - http://www.annauniv.edu/shan
Proc. Symp. Geometric Function Theory, Dec 2002, pp. 9–19.
GENERALISATION OF PICK-JULIA THEOREMS FOR OPERATORS IN A HILBERT SPACE RAJALAKSHMI RAJAGOPAL
Abstract. T. Ando and KyFan in their note ”Pick-Julia theorems for operators”, Math Z., 168, 23-34 (1979) have obtained Pick-Julia theorems for operator valued analytic functions. These results are generalised in this paper. Keywords: Operators, contraction, operator valued analytic function. AMS Subject Classification(2000): Primary 30C45.
1. Introduction Let H be a complex Hilbert space and T , a bounded linear transformation on H, which will be called an operator through out this paper. The norm of T denoted by k T k is given by k T k= sup{k T (x) k , x ∈ H, k x k= 1}. Let BL(H) denote the set of all bounded linear transformations or operators on H.A ∈ BL(H) is said to be invertible if there exists B ∈ BL(H) such that AB = BA = I, this bounded operator B, which is clearly unique is called the inverse of A and will be denoted by A−1 . An operator A is said to be self adjoint or Hermition if A = A∗ . An operator A is said to be unitary if AA∗ = A∗ A = I. For two Hermition operators A, B ∈ BL(H) we write A ≥ B to indicate A − Bis non negative, that is < (A − B)x, x >≥ 0, for every x ∈ H.A > B will mean that A − B is positive and invertible. For a positive operator A on H, the operator B on H such that BB = A is called the square root of A and denoted by A1/2 9
10
RAJALAKSHMI RAJAGOPAL
A contraction is an operator A with kAk ≤ 1, a proper contraction is an operator with kAk < 1. Potapav [4] has proved that (1)
kAk < 1 <=> I − A∗ A > 0.
Let f be a complex valued function which is analytic in a open neighbourhood N of the spectrum σ(T ) of an operator T , in the complex plane and let f (T ) denote the operator on H, defined by the Riesz Dunford integral [2] Z 1 f (T ) = f (z)(zI − T )−1 dz, 2πi c where I stands for the identity operator on H, C is a positively oriented simple closed rectifiable contour such that the inside domain D of C consists of σ(T ) and CU D ⊂ N . The limit used in defining the integral is in the norm topology of operators or uniform topology of operators. f (T ) can also be defined by the infinite series. f (T ) =
∞ X f (n) (0)
n!
n=0
Tn
which converge in the norm topology. Let F (T ) denote the family of all functions of f which are analytic on some neighbourhood of σ(T ) [The neighbourhood need not be connected and can depend on f ∈ F (T )]. Remark: All functions f of the form ∞ X αk λk f (λ) = k=0
valid in a neighbourhood of σ(T ) will be called a functional calculus for an operator T if the series ∞ X f (T ) = αk T k k=0
can be manipulated as if T were a number. Let E = {z :| z |<|}. Let D = {w :| w − η |< δ} where η =
1 − αβ 1− | β |2
δ =
|α−β | , 1− | β |2
and
PICK-JULIA THEOREMS
11
where α, β are complex numbers with | α |< 1, | β |< 1, | β |<| α | and 1 − Re(αβ) >| α − β |. Now D denote the image of E under φ where 1 + αz (2) φ(z) = 1 + βz Let z−1 ψ(z) = (3) ,z ∈ D α − βz clearly φ and ψ are inverses of each other and φ(E) = D and ψ(D) = E. We proceed to prove the following preliminary lemma, used in the sequel. Lemma 1. The Transformation δ(z − w) W = 2 δ − ((w − η)(z − η)) where w is a fixed point satisfying | w − η |< δ transforms D =| z − η |< δ onto | W |< 1, where (4) Proof
δ=
|α−β | 1 − αβ ;η = ; | α |< 1, | β |< 1 | 2 1− | β | 1− | β |2
Now W =
δ(z − w) δ 2 − (w − η)(z − η)
Then 0 z − w0 z−η 0 w−η 0 ,w = . | W |= 0 0 , where z = 1−wz δ δ 2 | z 0 − w0 | = | z 0 |2 + | w0 |2 −2Re(z 0 w0 ) We know | 1 − w0 z 0 |2 = 1+ | w0 |2 | z 0 |2 −2Re(z 0 w0 ) Therefore | z 0 − w0 | <| 1 − w0 z 0 | if and only if | z 0 |2 (1− | w0 |)2 < (1− | w0 |2 ). Thus, provided | w0 |< 1,the inequality holds if and only if | z 0 |< 1. That is, | W |< 1 if and only if | z − η |< δ, given that | w − η |< δ. Let H(E) denote the vector space of all analytic functions on E. For any two regions C1 and C2 of the complex plane, let H(C1 , C2 ) denote the class of all complex valued functions f analytic on C1 for which f (C1 ) ⊂ C2 . Let EH denote the set of all proper contraction on the
12
RAJALAKSHMI RAJAGOPAL
Hilbert Space H, that is all operators A on H such that k A k< 1. Let DH denote the set of all operators A on H with k A − ηI k< δ, where η and δ are given by (4). Let H(C, CH ) denote the class of operator valued functions F analytic in the region C such that F (z) ∈ CH where C is either E or D and CH will be either EH or DH . The next theorem due to Ky Fan [3] serves as an important tool in the proof of our results. Theorem A[3] For a complex valued function f ∈ H(E, E) and A ∈ EH , f (A) ∈ EH which in turn implies k f (A) k< 1.
2. Generalisation of Pick-Julia theorems for operator valued analytic functions. We require the following two lemmas to prove our theorems. Let (5) Φ(A, B) = (I − BB ∗ )−1/2 (A − B)(I − B ∗ A)−1 (I − B ∗ B)1/2 and −1/2 1 ∗ Ψ(A, B) = I − 2 (B − ηI)(B − ηI) (A − B) × δ ( −1 1/2 ) 1 ∗ (B ∗ − ηI) (A − ηI) −1 δ × I − 2 (B − ηI)(B − η1) . I− δ δ δ
(6) We now prove Lemma 2. Let A ∈ DH , B ∈ DH , then Ψ(A, B) ∈ EH ; and for any C ∈ EH , the inequality Ψ∗ (A, B)Ψ(A, B) ≤ C ∗ C is equivalent to −1 (B ∗ − ηI)(A − ηI) (A∗ − ηI)(A − ηI) (A∗ − ηI)(B − ηI) I− I− I− δ2 δ2 δ2 1/2 1/2 1 ∗ 1 ∗ −1 ∗ ≤ I − 2(7) (B − ηI)(B − ηI) [I − C C] I − 2 (B − ηI)(B − ηI) δ δ Proof (8)
Let A1 =
(A − ηI) (B − ηI) , B1 = δ δ
PICK-JULIA THEOREMS
13
Then k A1 k< 1, k B1 k< 1, since A ∈ DH , B ∈ DH . We can put the above in equality(7) in the form (9) (I−B1∗ A1 )(I−A∗1 A1 )−1 (I−A∗1 B1 ) ≤ (I−B1∗ B1 )1/2 (I−C ∗ C)−1 (I−B1∗ B1 )1/2 Now we have Ψ(A, B) = (I − B1 B1∗ )−1/2 (A1 − B1 )(I − B1∗ A1 )−1 (I − B1∗ B1 )1/2 So that Ψ∗ (A, B) = (I − B1∗ B1 )1/2 (I − A∗1 B1 )−1 (A∗1 − B1∗ )(I − B1 B1∗ )−1/2 Then I − Ψ∗ (A, B)Ψ(A, B) = I − (I − B1∗ B1 )1/2 (I − A∗1 B1 )−1 (A∗1 − B1∗ ) × (I − B1 B1∗ )−1 (A1 − B1 )(I − B1∗ A1 )−1 (I − B1∗ B1 )1/2 = (I − B1∗ B1 )1/2 (I − A∗1 B1 )−1 {(I − A∗1 B1 )(I − B1∗ B1 )−1 × (I − B1∗ A1 ) − (A∗1 − B1∗ )(I − B1 B1∗ )−1 (A1 − B1 )} × (I − B1∗ A1 )−1 (I − B1∗ B1 )1/2
(10) Consider
[(I − A∗1 B1 )(I − B1∗ B1 )−1 (I − B1∗ A1 ) − (A∗1 − B1∗ )(I − B1 B1∗ )−1 (A1 − B1 )] ∞ ∞ X X (B1 B1∗ )n (A1 − B1 )] (B1∗ B1 )n (I − B1∗ A1 ) − (A∗1 − B1∗ ) = [(I − A∗1 B1 ) n=0
n=0
=
∞ X
(B1∗ B1 )n
n=0
−
B1∗
∞ X n=0
(11)
(B1 B1∗ )n B1
+
A∗1
∞ X
(B1 B1∗ )n A1
n=0
−
A∗1
∞ X
(B1 B1∗ )n A1
n=0
= I + Σ(B1∗ B1 )n − Σ(B1∗ B1 )n − A∗1 IA = I − A∗1 A1
where all the series converge in the norm topology. Using (10) and (11) we obtain I − Ψ∗ (A, B)Ψ(A, B) = (I − B1∗ B1 )1/2 (I − A∗1 B1 )−1 (I − A∗1 A1 ) × (I − B1∗ A1 )−1 (I − B1∗ B1 )1/2 >0 since k B1 k < 1, k A1 k < 1, which implies that k Ψ(AB) k< 1, by (1). That is Ψ(A, B) ∈ EH . Now we observe that Ψ∗ (A, B)Ψ(A, B) ≤ C ∗ C is equivalent to I − Ψ∗ (A, B)Ψ(A, B) ≥ I − C ∗ C.This, in turn, is equivalent to (I − B1∗ B1 )−1/2 (I − B1∗ B1 )1/2 (I − A∗1 B1 )−1 (I − A∗1 A1 )(I − B1∗ A1 )−1 ≥ (I − B1∗ B1 )−1/2 (I − C ∗ C)(I − B1∗ B1 )−1/2
14
RAJALAKSHMI RAJAGOPAL
or (I − A∗1 B1 )−1 (I − A∗1 A1 )(I − B1∗ A1 )−1 ≥ (I − B1∗ B1 )−1/2 (I − C ∗ C)(I − B1∗ B1 )−1/2 . (I − B1∗ A1 )(I − A∗1 A1 )−1 (I − A∗1 B1 ) ≤ (I − B1∗ B1 )1/2 (I − C ∗ C)−1 (I − B1∗ B1 )1/2 Note that (I − C ∗ C)−1 exists because C ∈ EH . This completes the proof. Lemma 3 Let A ∈ DH , B ∈ DH and let C = ψ(A) and D = ψ(B) where ψ is given by (3). Thus C ∈ EH and D ∈ EH . Furthermore with Φ and Ψ given by (5) and (6) respectively, we have (12)
Φ∗ (C, D)Φ(C, D) = U ∗ Ψ∗ (A, B)Ψ(A, B)U
where U = S −1/2 (αI − βB ∗ ){(αI − βB ∗ )−1 S(αI − βB)−1 } is unitary and S = (| α |2 −1)I − (I− | β |2 )B ∗ B + 2ReB(1 − αβ). Proof A ∈ DH implies kA − ηIk < 1. δ Hence from (1) we have (A∗ − ηI) (A − ηI) > 0, δ δ that is, I − A∗1 A1 > 0, using (8). Let ηA = B + iC and ηA∗ = B − iC. Then B = ReηA where η, δ are given by (4). Now I - A∗1 A > 0 simplifies to I−
(13)
(| α |2 −1)I − A∗ A(1− | β |2 ) + 2ReA(1 − αβ) > 0
R 1 We define ψ(A) using (3) and the Dunford integral ψ(A) = 2πi ψ(z)(zI− c A)−1 dz so that ψ(A) = (A − I)(αI − βA)−1 = (αI − βA)−1 (A − I) and (14) ψ ∗ (A) = (αI − βA∗ )(A∗ − I) = (A∗ − I)(αI − βA∗ )−1 . To prove C = ψ(A) ∈ EH , let us consider I − Ψ∗ (A)Ψ(A) = I − (αI − βA)−1 (A∗ − I)(A − I)(αI − βA)−1 = (αI − βA∗ )−1 {(αI − βA∗ )(αI − βA) − (A∗ − I)(A − I)}(αI − βA)−1 = (αI − βA∗ )−1 {(| α |2 −1)I + 2ReA(1 − αβ) + A∗ A(| β |2 −1)}(αI − βA)−1
PICK-JULIA THEOREMS
15
I − Ψ∗ (A)Ψ(A) > 0 if and only if (| α |2 −1)I + 2ReA(1 − αβ) + A∗ (| β |2 −1) > 0 which is true by (13) Hence C ∈ EH . Similarly we can prove D ∈ EH Now D = ψ(B) = (αI − βB)−1 (B − I) Φ(C, D) = (I − DD∗ )−1/2 (C − D)(I − D∗ C)−1 (I − D∗ D)1/2 by (5). Consider
(I − DD∗ ) = I − (αI − βB)−1 (B − I)(B ∗ − I)(αI − βB ∗ )−1 = (αI − βB)−1 I(| α |2 −1) + BB ∗ (| β |2 −1) + 2ReB(1 − αβ) (αI − βB ∗ )− Now consider (C − D) = (αI − βA)−1 (A − I) − (αI − βB)−1 (B − I) = (αI − βB)−1 (A − B)(α − β)(αI − βA)−1 (I − D∗ C) = I − (αI − βB ∗ )−1 (B ∗ − I)(A − I)(αI − βA)−1 = (αI − βB ∗ )−1 [(αI − βB ∗ )(αI − βA) − (B ∗ − I)(A − I)](αI − βA)−1 = (αI − βB ∗ )−1 [(| α |2 −1)I − (1− | β |2 )B ∗ A +B ∗ (1 − βα) + A(1 − αβ)](αI − BA)−1 −1
(I − D∗ D) = I − (αI − βB ∗ ) (B ∗ − I)(B − I)(αI − βB)−1 −1
= (αI − βB ∗ ) [(αI − βB ∗ )(αI − βB) − (B ∗ − I)(B − I)](αI − βB)−1 . = (αI − βB ∗ )−1 [(| α |2 −1)I − (1− | β |2 )B ∗ B + 2ReB(1 − αβ)](αI − βB)−1 Let P = I(| α |2 −1) + BB ∗ (| β |2 −1) + 2ReB(1 − αβ). Q=A−B (15) R = I(| α |2 −1) − (1− | β |2 )B ∗ A + B ∗ (1 − βA) + A(1 − αβ). S = I(| α |2 −1) − (1− | β |2 )B ∗ B + 2ReB(1 − αβ)]. Now (16)
P = P ∗, S = S ∗.
LetA11 = (αI − βA); B11 = (αI − βB). Then using (15), Φ(C, D) can be written as −1 ∗−1 −1/2 −1 ∗−1 −1 −1 ∗−1 −1 1/2 Φ(C, D) = (B11 P B11 ) (α − β)(B11 QA−1 11 )(B11 RA11 ) (B11 SB11 ) ∗−1 ∗ −1 1/2 ∗ −1 ∗−1 ∗ ∗−1 ∗−1 ∗ ∗−1 −1/2 Φ∗ (C, D) = (B11 S B11 ) (A∗−1 . 11 R B11 )(A11 Q B11 )(α − β)(B11 P B11 )
16
RAJALAKSHMI RAJAGOPAL
Hence using(16) we obtain ∗−1 ∗ −1 1/2 ∗ ∗−1 −1 ∗−1 ∗ ∗−1 Φ∗ (C, D)φ(C, D) = (B11 S B11 ) (A∗−1 11 R B11 ) (A11 Q B11 ) −1 ∗ ∗−1 −1/2 −1 ∗−1 −1/2 (α − β)(B11 P B11 ) (B11 P B11 ) (α − β) −1 1/2 ∗−1 −1 −1 ∗−1 −1 QA−1 (B11 11 )(B11 RA11 ) (B11 SB11 ) ∗−1 ∗ −1 1/2 ∗ ∗−1 = (B11 S B11 ) (B11 R∗−1 A∗11 )(A∗−1 11 Q B11 )(α − β)(α − β) ∗−1 −1/2 −1 −1 ∗ −1 −1 ∗ ∗−1 −1/2 P B11 ) (B11 QA−1 (B11 P B11 ) (B11 11 )(A11 R B11 ) −1 1/2 ∗−1 ) SB11 (B11 ∗−1 ∗ −1 1/2 ∗−1 = (B11 S B11 ) (B11 R∗−1 Q∗ B11 ) −1 ∗ ∗−1 −1/2 (| α − β |)2 (B11 P B11 ) −1 ∗−1 −1/2 −1 ∗ (B11 P B11 ) (B11 QR−1 B11 ) −1 1/2 ∗−1 ) SB11 (B11
Now P = P ∗ and S = S ∗ ∗−1 −1 1/2 ∗ ∗−1 −1 1/2 = | α − β |2 (B11 SB11 (17) ) (B11 R∗−1 Q∗ P −1 QR−1 B11 )(B11 SB11 )
Next we have by (7) 1 (B − ηI)(B ∗ − ηI)}−1/2 δ(A − B)(δ 2 I − (B ∗ − ηI)(A − ηI)}−1 δ2 1 {I − 2 (B ∗ − ηI)(B − ηI)}1/2 δ where η, δ are given by (4). Consider 1 I − 2 (B − ηI)(B ∗ − ηI) δ (1 − αβ) | 1 − αβ |2 (1− | β |2 )2 ∗ BB − 2ReB + I = I− | α − β |2 (1− | β |2 ) 1− | β |2 1− | β |2 2 ∗ 2 = (| α | −1)I − BB (1− | β | ) + 2ReB(1 − αβ) | α − β |2
Ψ(A, B) = {I −
Therefore 1 (B − ηI)(B ∗ − ηI) 2 δ = (1− | β |2 )−1/2 | α − β |
I −
[I(| α |2 −1) − BB ∗ (1− | β |2 ) + 2ReB(1 − αβ)]−1/2 Since δ(A − B) =
|α−β | (A − B) 1− | β |2
PICK-JULIA THEOREMS
17
we have δ 2 I − (B ∗ − ηI)(A − ηI) = δ 2 I − (B ∗ A − ηAI − ηB ∗ I + ηηI) | α − β |2 1 − αβ (1 − βα)B ∗ | 1 − αβ |2 ∗ = I − B A− A− + I 1− | β |2 1− | β |2 1− | β |2 1− | β |2 1 ∗ 2 ∗ 2 αβ)A + (1 − αβ)B = (| α | −1)I − B A(1− | β | ) + (1 − 1− | β |2 [δ 2 I − (B ∗ − ηI)(A − ηI)]−1 = (1− | β |2 )[(| α |2 −1)I − B ∗ A(1− | β |2 ) +(1 − αβ)A + (1 − αβ)B ∗ ] 1 ∗ 1 ∗ ηI)(B − ηI) = I − (B − [B B − ηBI − ηB ∗ I − ηηI] δ2 δ2 (1− | β |2 )2 1 − αβ ∗ (1 − αβ)2 1 − αβ ∗ = I− BI − B I+ I B B− | α − β |2 1− | β |2 1− | β |2 1− | β |2 1 [(| α |2 −1)(1− | β |2 )I − B ∗ B(1− | β |2 )2 + (1 − αβ)(1− | β |2 )BI = | α − β |2
I−
+(1 − αβ)(1− | β |2 )B ∗ − | (1 − αβ) |2 I] Therefore [I −
1 ∗ (1− | β |2 )1/2 1/2 (B − [(| α |2 −1)I − B ∗ B(1− | β |2 ) ηI)(B − ηI)] = δ2 |α−β | +(1 − αβ)B + (1 − αβ)B ∗ ]1/2
So Ψ(A, B) =| α − β | (P −1/2 QR−1 S 1/2 ) Ψ(A, B) =| α − β | (S ∗1/2 R−1 Q∗ P ∗−1/2 ). Now P = P ∗ , S = S ∗ (18) Ψ∗ (A, B)Ψ(A, B) =| α − β |2 S ∗1/2 R∗−1 Q∗ P −1 QR−1 S 1/2 Where P, Q, R, S are given by (15). Let ∗ ∗−1 ∗ −1 1/2 U = S ∗−1/2 B11 (B11 S B11 ) ∗−1 −1 −1/2 U = (B11 SB11 ) B11 S −1/2 ∗−1 −1 1/2 ) B11 S −1/2 S ∗1/2 R∗−1 Q∗ U ∗ Ψ∗ (A, B)Ψ(A, B)U = | α − β |2 (B11 SB11 ∗−1 ∗ −1 1/2 ∗ (B11 S B11 ) P ∗−1/2 P 1/2 QR−1 S 1/2 S ∗−1/2 B11 ∗ = Φ (C, D)Φ(C, D) by (17) and (18).
18
RAJALAKSHMI RAJAGOPAL
Now it remains to prove that U is unitary operator. To do this, let us consider ∗ ∗−1 ∗ −1 1/2 ∗−1 −1 1/2 U U ∗ = S ∗−1/2 B11 (B11 S B11 ) (B11 SB11 ) B11 S −1/2 S ∗−1/2 ∗−1 ∗ −1 ∗ S B11 B11 S −1/2 B11 B11
= S ∗−1/2 S ∗ S −1/2 (S = S ∗ ) = I Similarly it can be shown that U ∗ U = I. This completes the proof of the lemma. We need the following theorem of T. Ando and KyFan for our further discussion. Theorem B[1] If F ∈ H(E, EH ) and z, w ∈ E, then z − w 2 I. Φ(F (z), F (w))Φ(F (z), F (w)) ≤ 1 − wz We proceed to prove the following Theorem 1. If G ∈ H(E, DH ) and z, w ∈ E, then z − w 2 ∗ I Ψ (G(z), G(w))Ψ(G(z), G(w)) ≤ 1 − wz Proof For z ∈ E, let F (z) = ψ(G(z)) when ψ is given by (3) . From lemma (2) it follows that ψ(G(z) ∈ EH . Hence F ∈ H(E, EH ). Also by (12). Φ∗ (F (z), F (w))Φ(F (z), F (w)) = U ∗ (W )Ψ∗ (G(z), G(w))Ψ(G(z), G(w))U (w) Where U (w) =
−1/2 (| α |2 −1)I − (1− | β |2 )G∗ (w)G(w) + 2ReG(w)(1 − αβ) n −1 [αI − βG∗ (w)] (αI − βG∗ (w)) (| α |2 −1)I − (1− | β |2 )G∗ (w)G(w) 1/2 +2ReG(w)(1 − αβ)(αI − βG(w))−1
is Unitary .Also F ∈ H(E, EH ). Hence by theorem 2 of T.Ando and KyFan We have U ∗ (w)Ψ∗ (G(z), G(w))Ψ(G(z), G(w))U (w) z−w ∗ IU (w), since U is unitary. ≤ U (w) 1 − wz This completes the proof.
PICK-JULIA THEOREMS
19
Theorem 2. If f ∈ H(D, EH ) and z, w ∈ D, then 2 δ(z − w) ∗ I Φ (f (z), f (w))Φ(f (z), f (w)) ≤ 2 δ − (w − η)(z − η) Proof Let F = f oφ where φ is given by (2). Then F ∈ H(E, EH ) Since f = F oψ, ψ being the inverse of φ, we have Φ(f (z), f (w)) = Φ[F (ψ(z)), F (ψ(w))] Hence by theorem 2 we get ψ(z) − ψ(w) 2 Φ∗ [F (ψ(z)), F (ψ(w))]Φ[F (ψ(z)), F (ψ(w))] ≤ I 1 − ψ(w)ψ(z) A direct but long calculation shows that ψ(z) − ψ(w) δ(z − w) = 2 1 − ψ(w)ψ(z) δ − (w − η(z − η)) and this completes the proof.Similar methods yield the following Theorem 3. If g ∈ H(D, DH ) and z, w ∈ D then 2 δ(z − w) I Ψ∗ (g(z), g(w))Ψ(g(z), g(w)) ≤ 2 δ − (w − η)(z − η) Remark By setting α = 1, β = −1 we deduce Pick-Julia theorems for operator valued analytic functions proved in [1]. ACKNOWLEDGEMENT The author is grateful to Prof. K S Padmanabhan for his fruitful suggestions during the preparation of this paper.
References [1] Ando T., and Ky Fan,Pick-Julia theorems for operators, Math.Z., 168, 23-34 (1979). [2] Dunford, N., and Schwartz, J.T.,Linear Operators I General theory, Wiley Interscience Press, New York, 1958. [3] Ky Fan, Analytic Functions of a proper contraction, Math. Z., 160 (1978), 275-290. [4] Potapav, V.P. The multiplicative structure of Contractive matrix functions, Amer.Math.Soc., Transl.(2) 15 (1960), 131-243.
20
RAJALAKSHMI RAJAGOPAL
Department of Mathematics, Loyola College, Chennai 600 034, India E-mail address:
[email protected]
Proc. Symp. Geometric Function Theory, Dec 2002, pp. 21–31
CONTRACTIONS OF SUBCLASSES OF HARMONIC UNIVALENT FUNCTIONS G. MURUGUSUNDARAMOORTHY, K. VIJAYA, AND THOMAS ROSY
Abstract. We consider complex valued harmonic functions of the form f = h + g where h and g are analytic in the open unit disc. For subclasses Goodman-Rønning type starlike and Goodman type convex harmonic functions, we investigate properties of the contraction map f (cz)/c, 0 < c < 1. For these classes, we determine extreme points, sharp coefficient and distortion bounds and various containment properties. Key words: contractions, harmonic, univalent, starlike, uniformly convex AMS Subject Classification: Primary 30 C45, Secondary 30C50
1. Introduction Recently, there has been interest[1,2,4,5,10] in studying the class of all complex valued harmonic functions defined in a simply connected domains. As is known such functions are representable in a simply connected domain D in the form f = h + g where h and g are analytic in this domain. Moreover, for mappings f of this type, the Jacobian Jf (z) = |h0 (z)|2 − |g 0 (z)|2 > 0,
z∈D
implies that a harmonic function f is sense preserving in D and locally univalent. Let H denote the family of functions f = h + g that are harmonic, complex valued, orientation preserving and univalent in the unit disk 21
22
G. MURUGUSUNDARAMOORTHY, K. VIJAYA, AND THOMAS ROSY
U = {z; |z| < 1}, with the normalization (1)
h(z) = z +
∞ X
n
an z , g(z) =
n=2
∞ X
bn z n
n=1
We call h the analytic and g the co-analytic part of f . Note that H reduces to the class S of nomalized analytic univalent functions whenever the co-analytic part of f is zero. We may sometimes restrict ourselves to the subclass H 0 of functions in H so that b1 = fz (0) = 0. It is shown in [2] that both H and H 0 are normal families while only H 0 is compact. Goodman [3] defined uniformly convex (UCV) function with the analytic characterization that φ ∈ UCV if and only if zφ00 (z) ζφ00 (z) Re 1 + 0 ≥ 0 , (z, ζ) ∈ U × U. φ (z) φ (z) Upon choosing ζ = −zeiλ in a suitable way, we may write φ ∈UCV if and only if (1 + eiλ )zφ00 (z) ≥ 0, (z, ζ) ∈ U × U. Re 1 + φ0 (z) . Ronning [8 ] defined the class Sp consisting of functions ψ(z) = zφ0 (z) such that φ ∈ U CV . Upon choosing ζ = zeiλ the analytic chracterization of Sp may be written as ψ ∈ U CV if and only if 0 iλ zψ (z) iλ Re (1 + e ) −e ≥ 0. ψ(z) Rosy and etal [6,7 ], generalized the classes Sp and U CV to include harmonic functions and defined the class 0 iλ zf (z) iλ GH (α) = f ∈ H : Re (1 + e ) −e >α f (z) (1 + eiλ )zf 00 (z) GKH (α) = f ∈ H : Re 1 + >α f 0 (z) where α(0 ≤ α < 1) and λ are real. We state the following theorems obtained in [ 6,7], which we shall need throughout this paper.
HARMONIC UNIVALENT FUNCTIONS
23
Theorem A[6] Let f = h + g so that h and g are given by (1). Then f is harmonic starlike of order α denoted by GH (α), if (2)
∞ X 2n − 1 − α n=2
1−α
|an | +
∞ X 2n + 1 + α n=1
1−α
|bn | ≤ 1,
0 ≤ α < 1.
The harmonic starlike functions f (z) = z +
(3)
∞ X n=2
where sharp.
P∞
n=2
|xn | +
∞ X 1−α 1−α n xn z + yn z n 2n − 1 − α 2n + 1 + α n=1
P∞
n=1
|yn | = 1, show that the bounds given by (2) is
Theorem B[7] Let f = h + g so that h and g are given by (1). Then f is harmonic convex of order α denoted by GKH (α), if (4)
∞ X n(2n − 1 − α)
1−α
n=2
|an | +
∞ X n(2n + 1 + α) n=1
1−α
|bn | ≤ 1,
0 ≤ α < 1.
The harmonic convex functions (5)
f (z) = z +
where sharp.
P∞
∞ X n=2
n=2
|xn | +
∞ X 1−α 1−α n xn z + yn z n n(2n − 1 − α) n(2n + 1 + α) n=1
P∞
n=1
|yn | = 1, show that the bounds given by (4) is
Motivated by Silverman[9], recently Ahuja and et. al. [1] studied the contractions of the mapping in harmonic starlike functions of order α and harmonic convex functions of order α in U . In this paper, we extend the above results to contractions of the mapping in GH (α) and GKH (α). A function F is said to be in GH (c, α) for some c, 0 ≤ c < 1 if F can be expressed by f (cz) h(cz) g(cz) = + , c c c f = h + g where h and g are the functions as given in (1) and satisfies the condition (2). Analogous to GH (c, α) is the family GKH (c, α) consisting of function F that can be expressed as (6), where f satis0 fies condition (4). Also let G0H (c, α) and GKH (c, α) be corresponding classes to GH (c, α) and GKH (c, α) where b1 = 0. (6)
F (z) =
24
G. MURUGUSUNDARAMOORTHY, K. VIJAYA, AND THOMAS ROSY
2. Main Results For f = h + g in Theorem A, it can be observed that the function F ∈ GH (c, α) may be written as F (z) = z +
(7)
∞ X
an c
n−1 n
z +
n=2
∞ X
bn cn−1 z n .
n=1
Hence if 0 < c1 ≤ c2 ≤ 1, then GH (c1 , α) ⊂ GH (c2 , α) ⊂ GH (α). 0 Theorem 1. G0H (c, α) ⊂ GKH (c, β) for (8) 1 h 1−β i n−1 inf2≤n<∞ 2n−1−α 1−α n(2n−1−β) c ≤ c0 = 1 h i n−1 1−β inf2≤n<∞ 2n+1+α 1−α n(2n+1+β)
if 0 ≤ α ≤ β < 1 if 0 ≤ β < α < 1
Theorem 1 is sharp for functions Fn (z) = z +
1−α 1−α cn−1 z n + cn−1 z n , (n = n(α, β)). 2n − 1 − α 2n + 1 + α
Proof. It suffices to show that ∞ X n(2n − 1 − β) n=2
≤
1−β
∞ X (2n − 1 − α) n=2
1−α
c
n−1
|an | +
∞ X n(2n + 1 + β)
1−β
n=1
|an | +
∞ X (2n + 1 + α) n=1
1−α
cn−1 |bn |
|bn | ≤ 1.
In view of (4) and (7) this holds if n(2n − 1 − β) n−1 (2n − 1 − α) c ≤ 1−β 1−α
and
n(2n + 1 + β) n−1 (2n + 1 + α) c ≤ 1−β 1−α
for each n. The required condition (8) is obtained by solving for c from these inequalities. As a consequence of the above theorem we have the following corollaries: 0 Corollary 1. G0H (c, α) ⊂ GKH (c, α) for 0 ≤ c < 1/3. 0 Corollary 2. G0H (c, α) ⊂ GKH (c, β) for 0 ≤ c <
1−α . 3−β
HARMONIC UNIVALENT FUNCTIONS
25
3. Extreme points and distortion bounds We now determine the extreme points of the compact family G0H (c, α) initially determininig the extreme points of the closed convex hull of G0H (1, α) denoted by clco G0H (1, α). Lemma 1. The extreme points of clco G0H (1, α) consists of functions h1 (z) = z,
hn (z) = z +
1−α eiαn z n , 2n − 1 − α
n = 2, 3, . . .
and gn (z) = z +
1−α eiβn z n , 2n + 1 + α
n = 1, 2, 3, . . . .
Proof. It is sufficient to prove that f ∈ clco GH (1, α) if and only if f can be expressed as f (z) =
∞ X
(λn hn (z) + µn gn (z))
n=1
where λn ≥ 0, µn ≥ 0 and f (z) =
n=1 (λn
+ µn ) = 1. Write
∞ X
∞ X (λn hn (z)+µn gn (z)) = z+
n=1
n=2
Then ∞ X 2n − 1 − α n=2
P∞
1−α
∞ X 1−α 1−α iαn n e λn z + eiβn µn z n . 2n − 1 − α 2n + 1 + α n=1
∞ X 2n + 1 + α 1 − α 1−α iαn |e |λn + |eiβn |µn 2n − 1 − α 1 − α 2n + 1 + α n=1
≤
∞ X n=2
λn +
∞ X
µn = 1 − λ 1 ≤ 1
n=1
and so f ∈ clco GH (1, α). Conversely suppose that f ∈ clco GH (1, α). From (2) it can be seen that 1−α 1−α |an | ≤ and |bn | ≤ 2n − 1 − α 2n + 1 + α for all n. Hence we may set λn =
2n − 1 − α 2n + 1 + α |an |(n = 2, 3, 4 . . .) and µn = |bn |(n = 1, 2, 3 . . .) 1−α 1−α
26
G. MURUGUSUNDARAMOORTHY, K. VIJAYA, AND THOMAS ROSY
and λ1 = 1 −
P∞
f (z) = z +
n=2 λn ∞ X
P∞
=
an z n +
2
= z+
∞ X n=2
= z+ = z+
∞ X n=2 ∞ X
n=1 µn . ∞ X
Then
bn z n
1 iαn n
|an |e
z +
∞ X n=1
∞ X 1−α 1−α iαn n e λn z + e−iβn µn z n 2n − 1 − α 2n + 1 + α n=1
λn (hn (z) − z) +
n=2
=
∞ X
|bn |e−iβn z n
∞ X
µn (gn (z) − z)
n=1
[λn hn (z) + µn gn (z)].
n=1
Theorem 2. The extreme points of G0H (c, α) consists of the functions 1−α P1 (z) = z, Pn (z) = z + xcn−1 z n , |x| = 1, (n = 2, 3, . . .) 2n − 1 − α and 1−α Qn (z) = z + ycn−1 z n , |y| = 1, (n = 1, 2, 3, . . .). 2n + 1 + α Proof. Consider the operator ∧ : G0H (1, α) → G0H (c, α) defined by ∧(f (z)) = F (z) = f (cz) . Note that clco G0H (1, α) = G0H (1, α) and ∧ is c an isomorphism from G0H (1, α) to G0H (c, α) and hence ∧ preserves extreme points and so the result can be seen to follow from Lemma 1. As the family G0H (c, α) is compact and convex, the maximum or minimum value of G0H (c, α) of the real part of any continuous linear functional occurs at one of the extreme points of G0H (c, α) and hence we have the following two corollaries: P P∞ n n 0 Corollary 3. If F (z) = z + ∞ n=1 An z + n=1 Bn z ∈ GH (c, α), then 1−α 1−α |An | ≤ cn−1 and |Bn | ≤ cn−1 . 2n − 1 − α 2n + 1 + α The bounds are sharp for 1−α 1−α F (z) = z + cn−1 z n + cn−1 z n ∈ G0H (c, α). 2n − 1 − α 2n + 1 + α
HARMONIC UNIVALENT FUNCTIONS
27
Corollary 4. If F (z) ∈ G0H (c, α), then r−
1−α 2 1−α 2 cr ≤ |F (z)| ≤ r + cr , 3+α 3−α
|z| = r < 1.
Equality holds for the functions F (z) = z +
1−α 2 1−α cz and F (z) = z + cz. 3−α 3+α
0 Theorem 3. The extreme points of GKH (c, α) consists of the functions
P1 (z) = z,
Pn (z) = z +
1−α xcn−1 z n , |x| = 1, (n = 2, 3, . . .) n(2n − 1 − α)
and 1−α ycn−1 z n , |y| = 1, (n = 1, 2, 3, . . .). n(2n + 1 + α) P∞ P n n 0 Corollary 5. If F (z) = z + ∞ n=1 Bn z ∈ GKH (c, α), n=2 An z + then 1−α 1−α |An | ≤ cn−1 and |Bn | ≤ cn−1 . n(2n − 1 − α) n(2n + 1 + α) Qn (z) = z +
The bounds are sharp for F (z) = z +
1−α 1−α 0 cn−1 z n + cn−1 z n ∈ GKH (c, α). n(2n − 1 − α) n(2n + 1 + α)
0 Corollary 6. If F (z) ∈ GKH (c, α), then
r−
1−α 1−α cr2 ≤ |F (z)| ≤ r + cr2 , 2(3 + α) 2(3 − α)
|z| = r < 1.
Equality holds for the functions F (z) = z +
1−α 1−α cz 2 and F (z) = z + cz 2 . 2(3 − α) 2(3 + α)
The distortion bounds for functions in GH (c, α) are discussed in the following theorem: 1−α Theorem 4. If F ∈ GH (c, α) and 1+α ≤ |b1 | < 1, then, for some 0 < c ≤ 1 and |z| = r < 1, 1−α 3+α 1−α 3+α 2 (1−|b1 |)r− − |b1 | cr ≤ |F (z)| ≤ (1+|b1 |)r+ − |b1 | cr2 3−α 3−α 3−α 3−α
28
G. MURUGUSUNDARAMOORTHY, K. VIJAYA, AND THOMAS ROSY
Proof. We prove only the right hand bound. The proof for the left hand bound is similar. Let F ∈ GH (c, α). Taking the absolute value of F , we have |F (z)| = (1 + |b1 |)r +
∞ X
(|an | + |bn |)cn−1 rn
n=2 ∞
1−αX ≤ (1 + |b1 |)r + 3 − α n=2
3−α 3+α |an | + |bn | cr2 1−α 1−α
∞ 1−α 2n + 1 + α 1−αX ≤ (1 + |b1 |)r + |an | + |bn | cr2 3 − α n=2 2n − 1 − α 1−α 1−α 3+α ≤ (1 + |b1 |)r + 1− |b1 | cr2 3−α 3−α 1−α 3+α ≤ (1 + |b1 |)r + − |b1 | cr2 . 3−α 3−α This bound is sharp for the function 1−α 3+α F (z) = z + |b1 |z + − |b1 | cz 2 . 3−α 3−α Similarly the distortion bounds for functions in GKH (c, α) can be determined. 1−α ≤ |b1 | < 1, then, for some Theorem 5. If F ∈ GKH (c, α) and 3+α 0 < c ≤ 1 and |z| = r < 1, 1 1−α 3+α 1 1−α 3+α 2 (1−|b1 |)r− − |b1 | cr ≤ |F (z)| ≤ (1+|b1 |)r+ − |b1 | cr2 . 2 3−α 3−α 2 3−α 3−α
Theorem 6. Let 0 < c ≤ 1 and 0 ≤ α < 1. Then G0H (c, α) ⊂ 0 GKH . The result is sharp with extremal func(c, β) if β = 3−α−3(1−α)c 3−α−(1−α)c tions 1 − α iθ 2 1 − α iφ 2 F (z) = z + e cz + e cz 3−α 3+α and F (z) = z +
1 − β iθ 2 1 − β iφ 2 e cz + e cz . 3−β 3+β
HARMONIC UNIVALENT FUNCTIONS
29
Proof. We need to show that if ∞ X 2n − 1 − α 2n + 1 + α |an | + |bn | ≤ 1 1 − α 1 − α n=2 then ∞ X 2n − 1 − β n=2
1−β
c
n−1
2n + 1 + β n−1 |an | + c |bn | ≤ 1. 1−β
It suffices to prove that both inequalities 1−α 1−β 1−α 1−β cn−1 ≤ , cn−1 ≤ 2n − 1 − α 2n − 1 − β 2n + 1 + α 2n + 1 + β holds for all integers n ≥ 2. Equivalently we must show for every n, that 2n − 1 − β 1 − α (9) η1 (n) = cn−1 ≤ 1, 1 − β 2n − 1 − α 2n + 1 + β 1 − α (10) cn−1 ≤ 1. η2 (n) = 1 − β 2n + 1 + α Since 3 − α 1 − α n−1 η1 (2) = c = 1, c(1 − α) 3 − α we need to show that if r1 (n) ≤ 1 and r1 (n + 1) ≤ 1; that is, if (2n − 1 − β)(1 − α)cn−1 ≤ (2n − 1 − α)(1 − β), then (2n + 1 − β)(1 − α)cn ≤ (2n + 1 − α)(1 − β). By induction (2n + 1 − β)(1 − α)cn = ≤ ≤ ≤
((2n − 1 − β)(1 − α)cn−1 c + 2(1 − α)cn (2n − 1 − α)(1 − β)c + 2(1 − α)cn (2n − 1 − α)(1 − β) + 2(1 − α)c2 (2n − 1 − α)(1 − β)
(1−α)c provided (1 − α)c2 ≤ (1 − β) = 3−α−(1−α)c or provided (1 − α)c2 − (3 − α)c + 1 ≥ 0 which holds if c ≤ 1. This proves (9).
Next to prove that (10) holds for all n and β ∗ = we note that β ∗ ≤ β because β − β∗ =
5+α−5(1−α)c . 5+α+(1−α)c
8(1 − α)2 c(c − 1) [(3 − α) − (1 − α)c][(5 + α) + (1 − α)c]
Initially
30
G. MURUGUSUNDARAMOORTHY, K. VIJAYA, AND THOMAS ROSY
for some c, (0 < c ≤ 1). Since η2 (2) = 1 for β = β ∗ , it is enough to prove that if (2n + 1 + β ∗ )(1 − α)cn−1 ≤ (2n + 1 + α)(1 − β ∗ ) , then (2n + 3 + β ∗ )(1 − α)cn ≤ (2n + 3 + α)(1 − β ∗ ). By induction (2n + 3 + β ∗ )(1 − α)cn = ≤ ≤ ≤
((2n + 1 + β ∗ )(1 − α)cn−1 c + 2(1 − α)cn (2n + 1 + α)(1 − β ∗ )c + 2(1 − α)cn (2n + 1 + α)(1 − β ∗ ) + 2(1 − α)c2 (2n + 3 + α)(1 − β)
6(1−α)c provided (1 − α)c2 ≤ (1 − β ∗ ) = (1−α)c+5+α which holds if (1 − α)c2 + (5 + α)c − 6 ≤ 0 and this inequality holds for c ≤ 1. This completes the proof.
Theorem 7. With the above notations, we have G0H (c, α) ⊂ G0H (c, β) with the extremal functions 1 − α iφ 2 1 − α iθ 2 e cz + e cz F (z) = z + 2(3 − α) 2(5 + α) and F (z) = z +
1 − β iφ 2 1 − β iθ 2 e cz + e cz . 2(3 − β) 2(5 + β)
References [1] O.P. Ahuja , J.M. Jahangiri and H. Silverman, Contractions of harmonic univalent functions Far. East. J. Math. Sci., 3(4), (2001), 691-704. [2] J. Clunie and T. Sheil - Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A.I. Math. 9 (1984), 3-25. [3] A.W Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), 86-92 [4] J.M. Jahangiri, Coefficient bounds and univalence criteria for harmonic functions with negative coefficients, Ann. Univ. Marie Curie - Sklodowska Sect. A 52 (1998) 57-66. [5] J.M. Jahangiri, Harmonic functions starlike in the unit disk, J. Math. Anal. Appl. 235 (1999) 303-309. [6] T. Rosy, B.A. Stephan, K.G. Subramanian and J.M. Jahangiri, GoodmanRønning type class of harmonic starlike functions, Kyungpook J. Maths. 41 (2001), 45-54. [7] T. Rosy, B.A. Stephan, K.G. Subramanian and J.M. Jahangiri, Goodman type harmonic convex functions, J. Natural Geo. (2002) 1-2.
HARMONIC UNIVALENT FUNCTIONS
31
[8] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 18 (1), (1993), 189-196. [9] H. Silverman, Contractions of starlike and convex mapping, Math. Japonica, 33 (1988) 303-309. [10] H. Silverman, Harmonic univalent functions with negative coefficients, J. Math. Anal. Appl. 220 (1998) 283 - 289.
Department of Mathematics, Vellore Institute of Technology, Deemed University, Vellore - 632 014 E-mail address:
[email protected]
Department of Mathematics, Vellore Institute of Technology, Deemed University, Vellore - 632 014
Department of Mathematics, Madras Christian College, Tambaram, Chennai - 600 059
Proc. Symp. Geometric Function Theory, Dec 2002, pp. 33–40
MEROMORPHIC STARLIKE FUNCTIONS WITH POSITIVE COEFFICIENTS ASSOCIATED WITH PARABOLIC REGIONS V RAVICHANDRAN, S SIVAPRASAD KUMAR, AND R USHA Abstract. We study two subclasses of meromorphic starlike functions with positive coefficients which are similar to classes related to uniformly convex functions. AMS Subject Classification: 30C45 Key Words: uniformly convex function, meromorphic starlike functions.
1. Introduction The class of uniformly convex functions introduced by Goodman[2] are studied by various authors. Several new classes of functions related to these two classes are also introduced and studied. Rønning[7] and Ma and Minda[5, 4] have given a one variable characterization of the class of uniformly convex functions. Also Rønning defined a class Sp of starlike functions and generalised this class to SP (α, β)[8]. Murugusudaramoorthy et. al.[9], Bharathi et. al.[1] and Kanas[3] have studied the class of functions called uniformly k-convex functions. In this paper, we study two subclasses of meromorphic starlike functions with positive coefficients which are similar to the above mentioned classes SP (α, β) and the uniformly k-convex functions. Let ∞ 1 X ∗ Σp = {f |f (z) = + an z n , an ≥ 0}. z 1 Let Σ∗p (β) is the subclass of Σ∗p consisting of starlike functions of order β. zf 0 (z) ∗ ∗ Σp (β) = f ∈ Σp | − Re ≥β . f (z) This class was studied by Mogra et. al. [6] and Uralegadi and Ganigi[10]. In this paper, we introduce two subclasses which are similar to the 33
34
V RAVICHANDRAN, S SIVAPRASAD KUMAR, AND R USHA
classes SP (α, β) and the class of k-convex functions and study their properties. 2. Main Results We begin with the following: Definition 1. Let 0 0 zf (z) ∗ zf (z) P M Sp (α, β) = f ∈ Σp | + α + β ≤ −Re +α−β . f (z) f (z) Definition 2. Let f ∈ Σ∗p satisfies 0 0 zf (z) zf (z) −Re ≥ α + 1 + β, f (z) f (z)
α ≥ 0,
0 ≤ β < 1.
The class of all such functions is denoted by P M KS(α, β) Theorem 1. If f (z) ∈ P M Sp (α, β), then ∞ X (n + β)an ≤ 1 − β, or equivalently f (z) ∈ Proof. Since f (z) =
1 z
n=1 Σ∗p (β).
+
P∞
n=1
an z n , we have ∞
−1 X nan z n zf (z) = + z n=1 0
and hence
P n+1 1− ∞ zf 0 (z) n=1 nan z P − . = ∞ f (z) 1 + 1 an z n+1 If f ∈ P M Sp (α, β), then P P∞ n+1 n+1 −1 + ∞ na z na z 1 − n n 1 1 1 + P∞ an z n+1 + α + β ≤ Re 1 + P∞ an z n+1 + α − β . 1 1 Let z → 1. Then we have P P −1 + ∞ 1− ∞ na n 1 1 nan 1 + P∞ an + α + β ≤ 1 + P∞ an + α − β 1 1 or ∞ ∞ ∞ ∞ X X X X nan + α + β + (α + β)an ≤ 1− nan +α−β+ (α−β)an −1 + n=1
n=1
n=1
n=1
or ∞ ∞ X X (n − α + β)an + 1 + α − β. (n + α + β)an + α + β − 1 ≤ − n=1
n=1
MEROMORPHIC STARLIKE FUNCTIONS
35
Therefore we have (since |x| ≤ n ⇔ −n ≤ x ≤ n) ∞ X
(n − α + β)an − 1 − α + β ≤
n=1
∞ X
(n + α + β)an + α + β − 1
n=1
≤−
∞ X
(n − α + β)an + 1 + α − β.
n=1
From the first two terms of the above inequalities, we have ∞ X −2α an ≤ 2α n=1
P∞
or − n=1 an ≤ 1, which is trivially true. Similarly from the other two terms, we have ∞ ∞ X X (n + α + β)an − 1 + α + β ≤ − (n − α + β)an + 1 + α − β n=1
or
n=1 ∞ X
2(n + β)an ≤ 2(1 − β)
n=1
or
P∞
∗ n=1 (n + β)an ≤ 1 − β or f ∈ Σp (β). This completes the proof.
Corollary 1. If f (z) ∈ P M Sp (α, β), then 1 1−β 1 1−β − r ≤ |f (z)| ≤ + r. r 1+β r 1+β The result is sharp for f (z) =
1 z
+
1−β z. 1+β
Corollary 2. If f (z) ∈ P M Sp (α, β), then 1−β an ≤ . n+β The result is sharp for f (z) =
.
1 z
+
1−β n z . n+β
To obtain a converse to Theorem 1, we note that if f ∈ Σ∗p (β), then zf 0 (z) − ≤1−β − 1 f (z)
This is a circular disk |w − 1| ≤ 1 − β. This disk should be contained in the parabolic region |w − (α + β)| ≤ Re(w + α − β) or equivalently in v 2 ≤ 4α(u − β) where w = −zf 0 (z)/f (z) = u + iv. Note that when α → 0, the parabolic region reduced to the line segment v = 0. Therefore, the class Σ∗p (β) 6⊆ P M Sp (α, β) for small α ≥ 0. Hence we
36
V RAVICHANDRAN, S SIVAPRASAD KUMAR, AND R USHA
determine conditions on α, β such that the disk |w −1| ≤ 1−β is inside |w − (α + β)| ≤ Re(w + α − β). To find such condition, we first find the radius of largest disk |w − a| ≤ Ra contained in the parabolic region. Then the required condition on α, β will be given by R1 ≥ 1 − β. The expression for Ra is given in the following: Lemma 1. : Let a > β. Then the disk |w − a| < Ra is contained in |w − (α + β)| ≤ Re{w + α − β} where
ap −β a ≤ 2α + β 2 α(a − α − β) a ≥ 2α + β
1p −β 1 ≤ 2α + β 2 α(a − α − β) 1 ≥ 2α + β
Ra = Note that R1 =
. If 2α+β ≥ p1, then R1 = 1−β ≥ 1−β. If 2α+β ≤ 1, then R1 ≥ 1−β provided 2 α(1 − α − β) ≥ 1 − β andp this is not the case since R1 obtained by taking minimum of 1 − β, 2 α(1 − α − β). Therefore we have the following. Corollary 3. Let α ≥
1−β . 2
Then
Σ∗p (β) ⊆ P M Sp (α, β). Corollary 4. For α ≥
1−β , 2
we have
Σ∗p (β) = P M Sp (α, β). Theorem 2. If f ∈ P M KS(α, β), then ∞ X
[(1 + α)n + (α + β)]an ≤ 1 − β
1
or equivalently f ∈ Σ∗p ( α+β ). 1+α Proof. Since for f ∈ P M KS(α, β), P 1− ∞ nan z n+1 zf 0 Pn=1 − = , n+1 f 1+ ∞ n=1 an z we have P P 1 − ∞ nan z n+1 1− ∞ nan z n+1 n=1 n=1 P∞ P∞ ≥ α − 1 + β. n+1 n+1 1 + n=1 an z 1 + n=1 an z
MEROMORPHIC STARLIKE FUNCTIONS
37
Let z → 1. Then we have 1− ⇔ ⇔
1−β− 1−β−
P∞
n=1
nan
P∞
n=1 (n
P∞
∞ ∞ X X an ] ≥ α (n + 1)an + β[1 + n=1 n=1 ∞ X ≥ α (n + 1)an
n=1 (n
+ β)an
≥
+ β)an
n=1 ∞ X
α(n + 1)an
n=1
P∞ ⇔ n=1 [α(n + 1) + (n + β)]an ≤ 1 − β P∞ ⇔ n=1 [(1 + α)n + (β + α)]an ≤ 1 − β P∞ β+α 1−β β+α ⇔ =1− ≤ n=1 (n + 1+α )an 1+α 1+α ∗ β+α ⇔ f ∈ Σp ( 1+α ) Corollary 5. If f (z) ∈ P M KS(α, β), then 1 1−β 1 1−β − r ≤ |f (z)| ≤ + r. r 1 + 2α + β r 1 + 2α + β The result is sharp for f (z) =
1 z
+
1−β z. 1+2α+β
Corollary 6. If f (z) ∈ P M KS(α, β), then 1−β an ≤ . (1 + α)n + α + β The result is sharp for f (z) =
1 z
+
1−β zn. (1+α)n+α+β
To obtain a converse to this theorem, we first determine the largest radius Ra such that the disk |w − a| ≤ Ra is contained in Rew ≥ α|w − 1| + β. Note that this inequality can be written as u−β 2 ( ) = (u − 1)2 + v 2 , w = u + iv. α The square of the distance from (a,0) to a point in Rew ≥ α|w − 1| + β is therefore given by u−β 2 d = (u − a)2 + ( ) − (u − 1)2 . α Since u−β d0 = 2[ 2 − a + 1], α
38
V RAVICHANDRAN, S SIVAPRASAD KUMAR, AND R USHA
we see that the minimum of d is given by u = α2 (a − 1) + β or u = α+β . 1+α α+β α+β 2 Also, if α (a − 1) + β ≤ 1+α the minimum is at u = 1+α . Otherwise it is the minimum of the two extremums. Note that α2 (a − 1) + β ≤
α+β (1 − β) ⇔a≤ + 1. 1+α α(1 + α)
1−β case (i). Let a ≤ 1 + α(1+α) . In this case, r α+β 1 α+β α+β Ra = ( − a)2 + 2 ( − β)2 − ( − 1)2 1+α α 1+α 1+α r α+β 1−β 2 β−1 2 = ( − a)2 + ( ) −( ) 1+α 1+α 1+α α + β ) = a − ( 1+α α+β = a−( ) 1+α n √ o (α+β) 1−β Case(ii). Let a ≥ 1 + α(1+α) . In this case, Ra = min a − (1+α) X where
X = (1 − a)2 − 2α2 (a − 1)2 + 2(β − 1)(1 − a) + α2 (a − 1)2 = (1 − a)2 (1 − α2 ) − 2(1 − β)(1 − a) = (1 − a)[(1 − a)(1 − α2 ) − 2(1 − β)]. Note that
Ra
(α + β) , = min a − (1 + α) (α + β) = min a − , (1 + α)
(1 − a)[(1 − a)(1 − α2 ) − 2(1 − β)] p 2 2 ((1 − a) (1 − α ) − 2(1 − β)(1 − a) p
We show that (−a + (α+β) )2 > (1 − a)2 (1 − α2 ) − 2(1 − β)(1 − a). Note (1+α) that the inequality is equivalent to (1 − a +
(α + β) − 1)2 > (1 − a)2 (1 − α2 ) − 2(1 − β)(1 − a). (1 + α)
Let 1 − a = x. Then we have (x +
β−1 2 ) > x2 (1 − α2 ) − 2(1 − β)x 1+α
MEROMORPHIC STARLIKE FUNCTIONS
39
or α2 x2 + 2x[
(β − 1) (β − 1) 2 1−β 2 + 1 − β] + ( ) = (αx + ) ≥ 0. 1+α 1+α 1+α
Lemma 2. If ( a − α+β a≤1+ Ra = p 1+α 2 2 ((1 − a) (1 − α ) − 2(1 − β)(1 − a) a > 1 +
(1−β) α(1+α) (1−β) α(1+α)
Then {w : |w − a| ≤ Ra } ⊆ {w : Rew ≥ α|w − 1| + β} Note that Ra < a − all a ≤ 1 +
(1−β) . α(1+α)
α+β 1+α
for all a ≥ 1 +
Hence if 1 ≤ 1 +
(1−β) α(1+α)
(1−β) α(1+α)
and Ra = a −
or 0 ≤
R1 = 1 −
(α + β) . (1 + α)
R1 ≥ 1 −
(α + β) . (1 + α)
(1−β) , α(1+α)
α+β 1+α
for
we have
Therefore
Hence we have the following: Theorem 3. Let f ∈ Σ∗p ( (α+β) ). Then (1+α) f ∈ P M KS(α, β). Corollary 1. P M KS(α, β) = Σ∗p (1 −
(α+β) ). (1+α)
References [1] R Bharathi, R Parvatham, A Swaminathan, On subclasses of uniformly convex functions and corresponding class of starlike functions, Tamkang J. Math.,28(1) (1997), 17–32. [2] A W Goodman, On uniformly convex functions, Ann. Polon. Math.56 (1991), no. 1, 87–92. [3] S Kanas, H M Srivastava Linear operators associated with k-uniformly convex functions. Integral Transform. Spec. Funct.9 (2000), no. 2, 121–132. [4] W Ma, D Minda, Uniformly convex functions. II. Ann. Polon. Math. 58 (1993), no. 3, 275–285. [5] W Ma, D Minda, Uniformly convex functions, Ann. Polon. Math. 57 (1992), no. 2, 165–175. [6] M L Mogra, T R Reddy, O P Juneja, Meromorphic univalent functions with positive coefficients, Bull. Austral. Math. Soc. 32 (1985), 161–176. [7] F Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc.118 (1993), no. 1, 189–196. [8] F Rønning, Integral representations of bounded starlike functions, Ann. Polon. Math. 60 (1995), no. 3, 289–297.
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V RAVICHANDRAN, S SIVAPRASAD KUMAR, AND R USHA
[9] K G Subramanian, G Murugusundaramoorthy, P Balasubrahmanyam, H Silverman, Subclasses of uniformly convex and uniformly starlike functions, Math. Japon. 42 (1995), no. 3, 517–522. [10] B A Uralegaddi, M D Ganigi, A certain class of meromorphically starlike functions with positive coefficients, Pure Appl. Math. Sci.26(1987), 75–81. Department of Mathematics, Sri Venkateswara College of Engineering, Sriperumbudur 602 105 E-mail address:
[email protected] Department of Mathematics, Sindhi College, 123, P.H.Road, Numbal, Chennai- 600 077 India E-mail address:
[email protected] Department of Mathematics, Sindhi College, 123, P.H.Road, Numbal, Chennai- 600 077 India
Proc. Symp. Geometric Function Theory, Dec 2002, pp. 41–48
CERTAIN SUBCLASSES OF MEROMORPHIC FUNCTIONS USING CONVOLUTION AND DIFFERENTIAL SUBORDINATION A GANGADHARAN
Abstract. In this paper, we define certain subclasses of meromorphic functions using convolution and subordination. We study some properties of these subclaseses. Key Words: Meromorphic functions, convolution, subordination, starlike and convex functions AMS Subject Classification: 30C45
1. Introduction Let M denote the set of all functions , analytic in the punctured unit disc (at the origin) and having a simple pole at the origin with residue unity. Let ∞ 1 X n an z , f (z) = + z 0 ∞
1 X n g(z) = + bn z , ∈ M . z 0 The convolution (Hadamard Product) of f and g is defined by ∞
1 X (f ∗ g)(z) = + an b n z n . z 0 For any f ∈ M, define Dα f (z) =
1 ∗ f (z) (α any real numz(1 − z)α+1
ber). Evidently, D0 f (z) = f (z) and D0 f (z) = zf 0 (z) + 2f (z). 41
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A GANGADHARAN
Also it is easy to verify that z(Dα f )0 (z) = (α + 1)Dα+1 f (z) − (α + 2)Dα f (z)
(1)
z(Dα f )0 (z) = Dα (zf 0 )(z).
(2)
and
Padmanabhan and Manjini (1968) used the notion of Hadamard product to define certain classes of meromorphic functions in the unit disk which reduce in special cases to the classes of convex, starlike meromorphic functions. Here in this paper, we generalize their classes and study their properties. We need the following lemma to prove our results: Lemma 1. [1] Let β, γ ∈ C, h ∈ H(U ) be convex univalent in U with h(0) = 1 and let q ∈ H(U ) with q(0) = 0 and Re(βq(z) + γ) > 0, z ∈ U . If p(z) = 1 + p1 z + . . . is analytic in U , then p(z) +
zp0 (z) ≺ h(z) ⇐ p(z) ≺ h(z). βq(z) + γ
2. Main Results Definition 1. Let g be any fixed function ∈ M. Let Mgk (h) denote the class of functions f ∈ M such that z(g ∗ f )00 (z) − 1+ ≺ h(z) (g ∗ f )0 (z) where (g ∗ f )0 (z) 6= 0 for z ∈ U. 1 1+z and h(z) = since (g ∗ f )(z) = f (z) z(1 − z) 1−z Mgk (h) reduces to the well known class of meromorphic convex univalent functions in U.
Remark 1. If g =
Definition 2. Let
∞
1 X f (z) = + an z n . z 0 Define hγ (z) =
∞ X γ+1 j=1
γ+j
z j−2
MEROMORPHIC FUNCTIONS
43
for Reγ > −1 and ∞ X γ+1 F (z) = (f ∗ hγ )(z) = ( )aj−1 z j−2 γ + j j=1
with a0 = 1, then γ+1 F (z) = γ+2 z
Z
z
tγ+1 f (t)dt.
0
Mgk (h)
and Re h is bounded in U then Theorem 1. Suppose f ∈ k F ∈ Mg (h) for Re(γ + 2) > δ provided (g ∗ F )0 (z) 6= 0 for z ∈ U, where F is as given in Definition 2. Proof. From the definition 2. we get, zF 0 (z) + (γ + 2)F (z) = (γ + 1)F (z) and so (g ∗ zF 0 )(z) + (γ + 2)(g ∗ F )(z) = (γ + 1)(g ∗ F )(z) using (1) z(g ∗ F )0 (z) + (γ + 2)(g ∗ F )(z) = (γ + 1)(g ∗ f )(z).
(3)
Differentiating (3) and letting −p(z) = 1 +
z(g ∗ F )00 (z) (g ∗ F )0 (z)
we get
−p(z) + (γ + 2) =
(γ + 1)(g ∗ f )0 (z) (g ∗ F )0 (z)
and so we have zp0 (z) z(g ∗ f )00 (z) p(z) + =− 1+ . −p(z) + (γ + 2) (g ∗ f )0 (z) Since f ∈ Mgk (h), we conclude by Lemma 1 that −p(z) ≺ h(z), for Re(−h(z) + γ + 2) > 0, z ∈ U which holds whenever Re(γ + 2) > δ. Hence F ∈ Mgk (h) for Re(γ + 2) > δ. Definition 3. Let Mgs (h) denote the class of functions f ∈ M such that (g ∗ f )0 (z) ≺ h(z) −z (g ∗ f )(z) where (g ∗ f )(z) 6= 0 for z ∈ U.
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A GANGADHARAN
1 1+z and h(z) = , then Mgs (h) reduces to the z 1−z well known class of meromorphic starlike univalent functions.
Remark 2. If g(z) =
Theorem 2. If f ∈ Mgk (h) then −zf 0 ∈ Mgs (h) Proof. Using (2) it is observed that z(g ∗ (−zf 0 ))0 (z) z(z ∗ f )00 (z) = 1 + (g ∗ (−zf 0 ))(z) (g ∗ f )0 (z) which implies the theorem.
Corollary 2. If f ∈ Mgk (h), Re h is bounded in U and g(z) = (γ + 2)F (z) − (γ + 1)f (z) where F is as given in definition 2 then g ∈ Mgs (h) for Re(γ + 2) > δ. Proof. Let f ∈ Mgk (h). By theorem 1, F ∈ Mgk (h) for Re(γ + 2) > δ. It follows from Theorem 2 that −zF 0 ∈ Mgs (h). But zf 0 (z) = (γ + 1)f (z) − (γ + 2)F (z). Hence the Corollary 2. k (h), (α any real number), denote the class of Definition 4. Let Mg,α all function f ∈ M such that (g ∗ f )00 (z) (g ∗ f )0 (z) Jg (α : f (z)) = α −1 − z + (1 − α) −z (g ∗ f )0 (z) (g ∗ f )(z)
with (g ∗ f 0 (z)) 6= 0 and (g ∗ f )(z) 6= 0 for z ∈ U. k (h) coincides with the class Mgk (h) Remark 3. For α = 1, the class Mg,α k and for α = 0 it reduces to the class Mgs (h). Thus the set Mg,α (h) gives a “continuous” passage from the class Mgk (h) to the class Mgs (h). k Theorem 3. (i) If f ∈ Mg,α (h) then k f ∈ Mg,0 (h) = Mgs (h) for α < 0. k k (ii) for α < β ≤ 0, Mg,α (h) ⊂ Mg,β (h).
Proof. (i) Let p(z) = −
z(g ∗ f )0 (z) , (g ∗ f )(z)
MEROMORPHIC FUNCTIONS
then −p(z) +
45
zp0 (z) z(g ∗ f )00 (z) =1+ p(z) g ∗ f 0 (z)
so it is seen that
αzp0 (z) + p(z). p(z) k (h) then by Lemma 1, it is concluded that p(z) ≺ h(z) If f ∈ Mg,α provided −h(z) Re( ) > 0, z ∈ U. α Jg (α, f (z)) = −
k (h) = Mgs (h) for α < 0. That is f ∈ Mg,0
(ii) If β = 0, Then this part reduces to part (i) therefore we assume k that β 6= 0. Suppose f ∈ Mg,α (h) then Jg (α; f (z)) ≺ h(z). Let z1 be any arbitrary point is U. Then Jg (α; f (z1 )) ∈ h(U ). Also by part (i) −z(g ∗ f )0 (z) ≺ h(z) (g ∗ f )(z) and so
−z1 (g ∗ f )0 (z1 ) ∈ h(U ). (g ∗ f )(z1 )
Now Jg (β : f (z)) = since 0 <
β 1− α
z(g ∗ f )0 (z) β − + Jα (α, f (z)) (g ∗ f )(z) α
β < 1 and h(U ) is convex, Jg (B; f (z)) ∈ h(U ). α
k Therefore Jg (β; f (z)) ≺ h(z) that is f ∈ Mg,β (h). k k (h). Mg,α (h) ⊆ f ∈ Mg,β
∞
1 X n Definition 5. Let g(z) = + bn z ). g −1 (z) is defined as z 0 g(z) ∗ g −1 (z) =
1 . z(1 − z)
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A GANGADHARAN
Then
∞
g −1 (z) =
1 X zn + . z b n 0
This function g −1 (z) is holomorphic in 0 < |z| < 1 with simple pole at z = 0. k Theorem 4. If f ∈ Mg,α (h) and if we choose that branch of α (g ∗ f )0 (z) −z (g ∗ f )(z)
which is equal to 1 at z = 0 then F ∈ Mgs1 (h) and G ∈ Mgs (h) where α (g ∗ f )0 (z) F (z) = (g ∗ f )(z) −z , (g ∗ f )(z) 1 . G(z) = g −1 (z) ∗ F (z) and g1 = z(1 − z) Proof. From the definition of F (z), it is seen that 1 F (z) = + A0 + A1 z + · · · z 0 −zF (z) (k) and = Jg (α, f (z)) ≺ h(z) since f ∈ Mg,α . So F ∈ Mgs1 (h). F (z) G ∈ Mgs (h) follows from the fact that −z(g ∗ G)0 (z) k = Jg (α; f (z))andf ∈ Mg,α (h). (g ∗ G)(z) Definition 6. If f ∈ Mgs (h) and k α = α(f ) = g.l.b [β/f ∈ Mg,β (h), β ≤ 0]
then f (z) is said to be of type α in Mgs (h) and written as f ∈ M (g, α). (Note that α is non-positive and may be −∞). Theorem 5. If f ∈ M (g, α) for some α in −∞ < α < 0 if and only k k if f ∈ Mg,β (h) for all β, α ≤ β ≤ 0 and f ∈ / Mg,β (h) for β < α. k Proof. If f ∈ M (g, α) then f ∈ Mg,β (h) for all β α < β ≤ 0. So Jg (β, f (z)) ≺ h(z) holds for z ∈ U and for all β, α < β ≤ 0. By ¯ ) for all z ∈ U where letting β → α we note that Jg (α; f (z)) lies in h(U
MEROMORPHIC FUNCTIONS
47
¯ ) is the closure of h(U ¯ ). Since Jg (α; f (z)) is an analytic function h(U in U by the Open mapping theorem it follows that the image of U by Jg (α; f (z)) must be a region or a point. But Jg (α; f (z)) is a nonconstant function because f (z) is so. Therefore the range of Jg (α; f (z)) must be a region and so Jg (α; f (z)) lies in h(U ) for all z ∈ U. That is k (h). The converse follows from the Jg (α; f (z)) ≺ h(z) hence f ∈ Mg,α definition of M (g, α). Definition 7. Let MgR (h) denote the class of functions f ∈ M such that z(g ∗ f )(z) ≺ h(z), z ∈ U. 1 1+z and h(z) = , then MgR (h) is the z(1 − z) 1−z class of functions such that Re(zf (z)) > 0.
Remark 4. If g = g1 =
Theorem 6. If f ∈ MgR (h) then F ∈ MgR (h) where F is given by Definition 2. Proof. Let p(z) = z(g ∗ F )(z) then p0 (z) = (g ∗ F )(z) + z(g ∗ F )0 (z). Using (1), zp0 (z) + p(z) = z(g ∗ f )(z) ≺ h(z), γ+1 Since f ∈ MgR (h). Using Lemma 1, it follows that F ∈ MgR (h) since Re(γ + 1) > 0. Definition 8. Let MgP (h) denote the class of functions f ∈ M such that −z 2 (g ∗ f )0 (z) ≺ h(z) z ∈ U. 1 1+z and h(z) = then MgP (h) is the z(1 − z) 1−z class of functions such that Remark 5. If g = g1 =
Re(z 2 f 0 (z)) > 0. Theorem 7. If f ∈ MgP (h) then F ∈ MgP (h) where F is given by definition 2. Proof. Let p(z) = −z 2 (g ∗ F )0 (z) Using the definition of F in (1), z(g ∗ F )0 (z) + (γ + 2)(g ∗ F )(z) = (γ + 1)(g ∗ f )(z) and so p(z) = (γ + 2)z(g ∗ F )(z) − (γ + 1)z(g ∗ f )(z)
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A GANGADHARAN
differentiating this with respect to z zp0 (z) + p(z) = −z 2 (g ∗ f )0 (z) ≺ h(z) γ+1 since, f ∈ MgP (h). Then by Lemma 1, F ∈ MgP (h).
Theorem 8. If f ∈ MgP (h) then −zf 0 ∈ MgR (h) . Proof. Since −z 2 (g ∗ f )0 (z) = z(g ∗ −zf 0 )(z) the result follows immediately. Theorem 9. The sets MgP (h) and MgR (h) are convex. Proof. Let f1 and f2 ∈ MgP (h) then −z 2 (g ∗ f1 )0 (z) ≺ h(z) and−z 2 (g ∗ f2 )0 (z) ≺ h(z). Let z1 be any arbitrary point in U. Then −z12 (g∗f1 )0 (z1 ) ∈ h(U ) and − z12 (g ∗ f2 )0 (z1 ) ∈ h(U ). Since h(U ) in convex, for 0 ≤ t ≤ 1, −tz12 (g ∗ f1 )0 (z1 ) − (1 − t)z12 (g ∗ f2 )0 (z1 ) ∈ h(U ). That is −z 2 [g ∗ (tf1 − (1 − t)f2 )0 ](z1 ) ∈ h(U ). Hence −z 2 [g ∗ (tf1 + (1 − t)f2 )0 ](z) ≺ h(z). This implies tf1 + (1 − t)f2 ∈ MgP (h). Thus MgP (h) is convex. Similarly MgR (h) is convex. References [1] K S Padmanabhan, R Parvatham, Some Applications of Differential Subordinations, Bull. Austral. Math. Soc. 32(1985), pp. 321–330. [2] S Padmanabhan, R Manjini, Certain Applications on Differential Subordinations, Pub. Inst. Mathematique, 39(53), pp/ 107-118.
Department of Applied Mathematics, Sri Venkateswara College of Engineering, Sriperumbudur 602 105 E-mail address:
[email protected]
Proc. Symp. Geometric Function Theory, Dec 2002, pp. 49–56
FUNCTIONS WITH NEGATIVE COEFFICIENTS DEFINED BY CONVOLUTION AND SUBORDINATION N MARIKKANNAN AND V RAVICHANDRAN
P∞ Abstract. Let g(z) = z + n=2 bn z n , bn > 0 be a fixed analytic function. In this paper, we study a class of functions f = P∞ ∗g)0 (z) z − n=2 an z n , an ≥ 0 where z(f (f ∗g)(z) ≺ h(z). Our results extend several earlier results. 2000 Mathematics Subject Classification 30C45 Key words: Starlike function, convolution, subordination, negative coefficients
1. Introduction and Definitions P n Let T be the class of all analytic functions f (z) = z − ∞ n=2 an z , an ≥ 0 where z ∈ ∆ = {z; |z| < 1}. A function f (z) ∈ T is called a function with negative coefficients. The subclass of T consisting of starlike functions of order α is denoted by T S ∗ (α) is studied by Silverman[2]. Several other class of starlike functions with negative coefficients were studied; for eg. see[1]. In fact, Ahuja [1] has studied the class Tλ (α) consisting of functions f (z) ∈ T satisfying the condition: z(Dλ f (z))0 Re > α, Dλ f (z)
z ∈ ∆, λ > −1, α < 1.
Since Dλ f (z) = f (z) ∗ (1−z)z λ+1 , we can write z(f ∗ g)0 (z) z Tλ (α) = f ∈ T ; > α, g(z) = , (f ∗ g)(z) (1 − z)λ+1 where * is Hadamard product of univalent functions. 49
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N MARIKKANNAN AND V RAVICHANDRAN
z Note that a function f (z) is convex if zf 0 (z) = f (z)∗ (1−z) 2 is starlike function. Therefore the subclass of convex functions can be defined as above with g(z) = z/(1 − z)2 . Also the function f is starlike of order α if zf 0 (z) 1 + (1 − 2α)z ≺ . f (z) 1−z The class of starlike functions can be extended to a class where we replace the particular function 1+(1−2α)z by a more general function 1−z h(z).
In this article, we study the following class of functions which gives several well-known subclasses of functions for suitable choices of g(z), h(z): P n Definition 1. Let g(z) = z + ∞ n=2 bn z be a fixed analytic function in ∆ = {z : |z| < 1} and bn > 0, (n ≥ 2). P The class Sg∗ (h) consists n of analytic functions of the form f (z) = z + ∞ n=2 an z where an ≥ 0 ,(n ≥ 2), which satisfies z(f ∗ g)0 (z) ≺ h(z), z ∈ ∆. (f ∗ g)(z) Let T Sg∗ (h) be the subclass of Sg∗ (h) consisting of functions with negative coefficients. When g(z) = z/(1 − z), h(z) = (1 + z)/(1 − z), the class T Sg∗ (h) is the class T S ∗ (α) of starlike functions with negative coeffcients of order α introduced and studied by Silverman[2]. Also when g(z) = z/(1 − z), Sg∗ (h) = S ∗ (h) and T Sg∗ (h) = T S ∗ (h) We assume that h(z) is an analytic function in ∆ with h(0) = 1, zh0 (z) h (0)h00 (0) > 0 and Reh(z) > α = h(−1). and Φ(z) = h(z)−1 is analytic univalent function in ∆ which maps ∆ onto a starlike region with respect to 1. 0
With this assumption on h, we have the following: Theorem 1. [3] A function f ∈ T S ∗ (h) if and only if ∞ X n=2
or f ∈ T S ∗ (α).
(n − α)|an | ≤ 1 − α
FUNCTIONS WITH NEGATIVE COEFFICIENTS
51
In this paper, we obtain the coefficient inequality, coefficient estimate, distortion theorem, and a convolution result for functions in our class T Sg∗ (α). Several earlier results follows from our results and are not indicated here.
2. Main Results We first prove a necessary and sufficient condition for functions to be in the class T Sg∗ (h) in the following: P n ∗ Theorem 2. A function f (z) = z − ∞ n=2 an z ∈ T Sg (h) if and only P∞ if n=2 (n − α)an bn ≤ 1 − α. Proof. Since f ∈ T Sg∗ (h) if and only if f ∗g ∈ T S ∗ (α), the result follows from Theorem 1. 1−α Theorem 3. If f ∈ T Sg∗ (h) then an ≤ (n−α)b with the equality only n 1−α n for functions of the form fn (z) = z − (n−α)bn z .
Proof. If f ∈ T Sg∗ (h), then (n − α)an bn ≤ an ≤ Clearly for fn (z) = z −
X
1−α . (n − α)bn
1−α zn (n−α)bn
an =
(n − α)an bn ≤ 1 − α
∈ T Sg∗ (h), we have
1−α . (n − α)bn
Theorem 4. If f ∈ T Sg∗ (h), then r−
1−α 2 1−α 2 r ≤ |f (z)| ≤ r + r , (2 − α)b2 (2 − α)b2
provided bn ≥ b2 . The result is sharp for f (z) = z +
1−α 2 z . (2 − α)b2
|z| = r < 1,
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N MARIKKANNAN AND V RAVICHANDRAN
Proof. Let |z| = r. Since f (z) = z −
P∞
n=2
|f (z)| ≤ r +
∞ X
an z n , we have
an r n
n=2
≤ r+
∞ X
an r 2
n=2
≤ r+r
2
∞ X
an .
n=2
Since bn ≥ b2 > 0, we note that (2 − α)b2 ≤ (n − α)bn . Therefore, from ∞ X
(n − α)an bn ≤ 1 − α,
n=2
we obtain b2 (2 − α)
∞ X
an ≤
n=2
∞ X
(n − α)an bn ≤ 1 − α
n=2
or ∞ X n=2
an ≤
1−α . (2 − α)b2
This shows that |f (z)| ≤ r + r2
1−α (2 − α)b2
|f (z)| ≥ r − r2
1−α . (2 − α)b2
and similarly we have
Since T Sg∗ (h) = T Sg∗ (α), we hereafter state our results in the latter notation.
FUNCTIONS WITH NEGATIVE COEFFICIENTS
Theorem 5. Let f ∈ T Sg∗1 (α), gi (z) = z − h(z) = z − f (z) = z −
∞ X n=2 ∞ X n=2 ∞ X
53
g ∈ T Sg∗2 (β) where
bin z n ,
bin ≥ 0 for i = 1, 2,
hn z n ,
hn ≥ 0,
fn z n ,
fn ≥ 0,
gn z n ,
gn ≥ 0,
n=2
g(z) = z −
∞ X n=2
then f ∗ g ∈
T Sh∗ (γ) γ=
where
(2 − α)(2 − β)b12 b22 − 2h2 (1 − α)(1 − β) (2 − α)(2 − β)b12 b22 − h2 (1 − α)(1 − β)
provided b1n b2n > nhn . Proof. Since f ∈ T Sg∗1 (α), we have ∞ X n−α
(1)
n=2
1−α
fn b1n ≤ 1.
Also g ∈ T Sg∗2 (β) implies ∞ X n−β
(2)
n=2
Now f ∗ g = z − (3)
P∞
n=2
1−β
gn b2n ≤ 1.
fn gn z n ∈ T Sh∗ (γ) if ∞ X n−γ n=2
1−γ
fn gn hn ≤ 1.
To prove (1) and (2) implies (3), we note that (4) ! 12 12 ∞ ∞ X X n−α (n − α)(n − β) fn gn b1n b2n ≤ fn b1n (1 − α)(1 − β) 1 − α n=2 n=2 To prove (3), it is enough to show that 12 n−γ (n − α)(n − β) fn gn hn ≤ fn gn b1n b2n 1−γ (1 − α)(1 − β)
∞ X n−β n=2
1−β
! 12 gn b2n
≤ 1.
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N MARIKKANNAN AND V RAVICHANDRAN
or equivalently, (5)
p
21 1−γ (n − α)(n − β) fn gn ≤ b1n b2n . (n − γ)hn (1 − α)(1 − β)
From (4), we have s p fn gn ≤
(6)
(1 − α)(1 − β) 1 . (n − α)(n − β) b1n b2n
In view of (5) and (6), it is enough to prove that s 12 (1 − α)(1 − β) 1 1−γ (n − α)(n − β) ≤ b1n b2n (n − α)(n − β) b1n b2n (n − γ)hn (1 − α)(1 − β) n−γ 1 ≤ 1−γ hn
⇔
s
(n − α)(n − β) b1n b2n (1 − α)(1 − β)
s
(n − α)(n − β) b1n b2n (1 − α)(1 − β)
n−γ 1 (n − α)(n − β) ≤ b1n b2n . 1−γ hn (1 − α)(1 − β)
⇔
Let B=
1 (n − α)(n − β) b1n b2n . hn (1 − α)(1 − β)
Then B ≥ n > 1 and n − γ ≤ B − Bγ. This gives γ≤
1− B−n = B−1 1−
n B 1 B
.
Therefore we have γ≤
1− 1−
nhn (1−α)(1−β) (n−α)(n−β)b1n b2n hn (1−α)(1−β) (n−α)(n−β)b1n b2n
= G(n).
Clearly, G(n) is an increasing function of n and hence G(n) ≥ G(2). Hence γ ≤
(2 − α)(2 − β)b12 b22 − 2h2 (1 − α)(1 − β) . (2 − α)(2 − β)b12 b22 − h2 (1 − α)(1 − β)
Hence the result follows.
FUNCTIONS WITH NEGATIVE COEFFICIENTS
55
P∞ P n n Theorem 6. If fP f2 (z) = z − ∞ 1 (z) = z − n=2 an z , n=2 cn z , ∞ ∗ 2 n 2 and f3 (z) = z − n=2 (an + cn )z where f1 (z), f2 (z) ∈ T Sg (α), then f3 ∈ T Sg∗ (β), where −3α2 + 4α . β≤ 2 − α2 Proof. Since f1 (z) ∈ T Sg∗ (α), we have ∞ X n−α n=2
and hence
1−α
an bn ≤ 1,
∞ X n−α 2 2 2 ( ) a b ≤ 1. 1−α n n n=2
Also since f2 (z) ∈ T Sg∗ (α), we have ∞ X n−α n=2
and hence
1−α
cn bn ≤ 1,
∞
1X n−α 2 2 2 ) c b ≤ 1. ( 2 n=2 1 − α n n Hence (7)
∞ X n−α 2 2 ) (an + c2n )b2n ≤ 1. ( 1 − α n=2
We have to show that ∞ X n−β n=2
1−β
(a2n + c2n )bn ≤ 1.
The last inequality follows if n−β 1 n−α 2 ≤ ( ). 1−β 2 1−α Solving for β, we have β≤
(n − α)2 − 2n(1 − α)2 . (n − α)2 − 2(1 − α)2
As the right hand side of the above inequality is increasing function of n, we get (2 − α)2 − 4(1 − α)2 β≤ (2 − α)2 − 2(1 − α)2
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N MARIKKANNAN AND V RAVICHANDRAN
or β≤
4α − 3α2 . 2 − α2
Hence the result follows. References
[1] O P Ahuja, Hadamard products of Analytic Functions Defined by Ruscheweyh Derivatives, in: Current Topics in Analytic Function Theory, (H M Srivastava, S Owa, editors,), World Scientific, Singapore, 1992, pp. 13-28. [2] H Silverman, Univalent Functions with Negative Coefficients, Proc. Amer. Math. Soc. 51(1975), pp. 109-116. [3] V Ravichandran, On Starlike Functions with Negative Coefficients, preprint.
Department of Applied Mathematics, Sri Venkateswara College of Engineering, Sriperumbudur 602 105
Department of Computer Applications, Sri Venkateswara College of Engineering, Sriperumbudur 602 105
Proc. Symp. Geometric Function Theory, Dec 2002, pp. 57–62
ON CERTAIN FUNCTIONS DEFINED BY RUSCHEWEYH DERIVATIVES V. RAVICHANDRAN, C SELVARAJ, AND RAJALAKSHMI RAJAGOPAL
Abstract. In this paper, we extend a result on sufficient condition for starlikeness of order β for functions defined using Ruscheweyh derivatives. Key Words: Sufficient conditions, Ruscheweyh derivatives Mathematics Subject Classification: 30C45
Dedicated to the memory of Prof K S Padmanabhan 1. Introduction Let An be the class of all functions f (z) = z + an+1 z n+1 + · · · which are analytic in 4 = {z; |z| < 1} and let A1 = A. A function f (z) ∈ A is starlike of order α, 0 ≤ α < 1 if 0 zf (z) >α Re f (z) for all z ∈ 4. The class of all starlike functions of order α is denoted by S ∗ (α). We write S ∗ (0) simply as S ∗ . Recently Li and Owa [2] have proved the following: Theorem 1. If f (z) ∈ A satisfies 0 zf (z) zf 00 (z) α Re α 0 +1 >− , f (z) f (z) 2
z∈4
for some α (α ≥ 0), then f (z) ∈ S ∗ . In fact, Lewandowski, Miller and Zlotkiewics [1] and Ramesha, Kumar, and Padmanabhan [4] have proved weaker form of the above Theorem. If the number −α/2 is replaced by −α2 (1 − α)/4, (0 ≤ α < 2) 57
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V. RAVICHANDRAN, C SELVARAJ, AND RAJALAKSHMI RAJAGOPAL
in the above condition, Li and Owa [2] have proved that f (z) is in S ∗ (α/2). Later the authors[5] have proved the following: Theorem 2. If f (z) ∈ An satisfies 0 zf (z) zf 00 (z) n αn Re α 0 +1 > αβ[β + − 1] + [β − ], f (z) f (z) 2 2
z ∈ 4,
(0 ≤ α, β ≤ 1), then f (z) ∈ S ∗ (β). In this paper we extend the above sufficient condition for functions defined by Ruscheweyh derivatives. To prove our result we need the following: Lemma 1. [3] Let Ω be a set in the complex plane C and suppose that Φ(z) is a mapping from C 2 × 4 to C which satisfies Φ(ix, y; z) 6∈ Ω for z ∈ 4, and for all real x, y such that y ≤ −n(1 + x2 )/2. If the function p(z) = 1 + cn z n + · · · is analytic in 4 and Φ(p(z), zp0 (z), z) ∈ Ω for all z ∈ 4, then Re p(z) > 0.
2. Main Result We begin with the definition of Ruscheweyh derivative: Definition 1 (2). Let f ∈ A. Then the Ruscheweyh derivative of f (z), denoted by Dδ f, is defined by Dδ f = f (z) ∗
z z , δ ≥ −1, where f (z) ∗ (1 − z)δ+1 (1 − z)δ+1
is the Hadamard product of f (z) and z/(1 − z)δ+1 . It is easy to verify the relation (1)
z(Dδ f (z))0 = (δ + 1)Dδ+1 f (z) − δDδ f (z).
We shall now prove a theorem which gives asufficient condition for δ+1 δ functions f (z) ∈ An to satisfy the condition Re D f (z)/D f (z) > β.
ON CERTAIN FUNCTIONS DEFINED BY RUSCHEWEYH DERIVATIVES 59
Theorem 1. Let f ∈ An and δ ≥ −1, 0 ≤ β < 1. If δ+2 Dδ+1 f (z) D f (z) Dδ+1 f (z) Re a δ+1 +b +c Dδ f (z) D f (z) Dδ f (z) n a(1 − β) δ+1 a 2 ≥− + a +b β + + c β, 2 δ+2 δ+2 (δ + 2)
where a > 0, b and c are real and 2[(δ + 1)a + (δ + 2)b](1 − β) + na ≥ 0, then Re
Dδ+1 f (z) Dδ f (z)
≥ β,
z ∈ 4.
Proof. Let p(z) be defined by
(1 − β)p(z) + β =
Dδ+1 f (z) . Dδ f (z)
Differentiating the above equation and using (1), we have z(Dδ+1 f (z))0 z(Dδ f (z))0 (1 − β)zp0 (z) = − (1 − β)p(z) + β Dδ + 1f (z) Dδ f (z) (δ + 2)Dδ+2 f (z) − (δ + 1)Dδ+1 f (z) = Dδ+1 f (z) (δ + 1)Dδ+1 f (z) − δDδ f (z) − Dδ f (z) (δ + 2)Dδ+2 f (z) (δ + 1)Dδ+1 f (z) = − δ − Dδ+1 f (z) Dδ f (z)
Therefore, we have
(δ + 2)
Dδ+2 f (z) (1 − β)zp0 (z) = 1 + + (δ + 1)((1 − β)p(z) + β). Dδ+1 f (z) (1 − β)p(z) + β
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Thus δ+2 Dδ+1 f (z) D f (z) Dδ+1 f (z) a δ+1 +b +c Dδ f (z) D f (z) Dδ f (z) a(1 − β) 2a(δ + 1) = + β(1 − β) b + + b + c(1 − β) p(z) δ+2 δ+2 a(δ + 1) a(1 − β) 0 2 + zp (z) + (1 − β) b + + b p(z)2 δ+2 δ+2 aβ a(δ + 1) 2 + +β b+ + cβ δ+2 δ+2 = Φ(p(z), zp0 (z); z) For all real x and y satisfying y ≤ −n/2(1 + x2 ), we have
Φ(ix, y; z) =
2a(δ + 1) a(1 − β) + β(1 − β) b + + b + c(1 − β) ix δ+2 δ+2 a(1 − β) a(δ + 1) 2 + y − (1 − β) b + + b x2 δ+2 δ+2 aβ a(δ + 1) 2 +β b+ + + cβ δ+2 δ+2
and therefore we have a(1 − β) a(δ + 1) 2 ReΦ(ix, y; z) = y − (1 − β) b + + b x2 δ+2 δ+2 a(δ + 1) aβ 2 +β b+ + cβ + δ+2 δ+2 n a(1 − β) n a(1 − β) a(δ + 1) 2 ≤ − − + (1 − β) b + x2 2 δ+2 2 δ+2 δ+2 a(δ + 1) aβ + β2 b + + cβ + δ+2 δ+2 n a(1 − β) aβ a(δ + 1) 2 + +β b+ + cβ. ≤ − 2 δ+2 δ+2 δ+2 The last inequality follows since n a(1 − β) a(δ + 1) 2 + (1 − β) b + ≥ 0. 2 δ+2 δ+2
ON CERTAIN FUNCTIONS DEFINED BY RUSCHEWEYH DERIVATIVES 61
n o a(δ+1) aβ 2 Define Ω = ω : Reω > − n2 a(1−β) b + + cβ . Then + + β δ+2 δ+2 δ+2 0 Φ(p(z), zp (z), z) ∈ Ω and Φ(ix, y; z) 6∈ Ω for all real x and y such that y ≤ −n(1 + x2 )/2, z ∈ E. By an applications of the above Lemma 1, the result follows.
By taking n = 1 and β = 0, we have the following: Corollary 1. Let f ∈ A and δ ≥ −1. If Re
Dδ+1 f (z) Dδ f (z)
Dδ+2 f (z) Dδ+1 f (z) a δ+1 +b +c D f (z) Dδ f (z)
≥−
a , 2(δ + 2)
where a > 0, b and c are real, and 2[(δ + 1)a + (δ + 2)b] + a ≥ 0, then Re
Dδ+1 f (z) Dδ f (z)
≥ 0.
Note that D0 f (z) = f (z), D1 f (z) = zf 0 (z) and D2 f (z) = zf 0 (z) + By taking δ = 0, a/2 = α, b = 0, a + c = 1, we have the Theorem 2. 1 2 00 z f (z). 2
References [1] Lewandowski, Zdzislaw; Miller, Sanford; Zlotkiewicz, Eligiusz. Generating functions for some classes of univalent functions. Proc. Amer. Math. Soc. 56 (1976), 111–117. [2] Lin, Li Jian; Owa, Shigeyoshi. Sufficient Conditions for Starlikeness. Indian J. Pure Appl. Math. 33(3) (2002), 313–318. [3] Miller, Sanford S.; Mocanu, Petru T. Differential subordinations and inequalities in the complex plane. J. Differential Equations 67(2) (1987), 199–211. [4] Ramesha, C.; Kumar, Sampath; Padmanabhan, K. S. A sufficient condition for starlikeness. Chinese J. Math. 23(2) (1995), 167– 171. [5] V. Ravichandran, S. Selvaraj, R. Rajalaksmi, Sufficient conditions for starlikeness of order β, J. inequal. pure & appl. math. (to appear). [6] St. Ruscheweyh, New criteria for univalent functions , Proc.Amer. Math., Soc. , 49 (1975), 109 -115.
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V. RAVICHANDRAN, C SELVARAJ, AND RAJALAKSHMI RAJAGOPAL
Department of Computer Applications, Sri Venkateswara College of Engineering, Pennalur 602 105, India E-mail address:
[email protected]
Department of Mathematics, L N Government College, Ponneri, 601 204, India
Department of Mathematics, Loyola College, Chennai 600 034
Proc. Symp. Geometric Function Theory, Dec 2002, pp. 63–68
A NOTE ON A SUBCLASS OF CLOSE-TO-CONVEX FUNCTIONS N. MAGESH, C. SELVARAJ, AND RAJALAKSHMI RAJAGOPAL
Abstract. In this paper a new subclass of close-to-convex functions is defined and its properties like co-efficient estimates, distortion theorems, radius of convexity and other properties are studied. Keywords: close-to-convex functions, convex functions, starlike functions;
AMS Subject Classification: 30C45
1. Introduction Let A denote the class of functions of the form (1)
f (z) = z +
∞ X
an z n ,
2
that are analytic in the unit disc E = {z; |z| < 1}. Let S denote the subclass A consisting of univalent functions in E. Let C denote the class of all convex function in E. 0
Definition 1. Let f ∈ A. Then f is said to belong to the class C (α) if there exists a convex function g such that (2)
Re
0
zf (z) g(z)
> α, 0 ≤ α < 1, z ∈ E. 63
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N. MAGESH, C. SELVARAJ, AND R RAJALAKSHMI
Theorem 1. (coefficient estimates.) ∞ P 0 Let f (z) = z + an z n ∈ C (α), |z| < 1. Then for z ∈ E, 2
|an | ≤ 2 −
(3)
1 f or all n ≥ 2, 0 ≤ α < 1. n
0
Proof. f ∈ C (α) implies 0
(4)
zf (z) = (1 − α)p(z) + α, where p(0) = 1, Re p(z) > 0. g(z)
Then 0
zf (z) = [(1 − α)p(z) + α]g(z). ∞ ∞ P P Let g(z) = z + bn z n ∈ C and let p(z) = 1 + cn z n , |z| < 1.
(5)
2
1
Then 0
zf (z) = (1 − α)(1 + c1 z + c2 z 2 + · · · + cn z n + . . . )(z + b2 z 2 + · · · + bn z n + . . . ) +α(z + b2 z 2 + · · · + bn z n + . . . ) Comparing the coefficients of z n on both sides, we get, (6)
nan = (1 − α) (cn−1 + b2 cn−2 + b3 cn−3 + .... + bn−1 c1 ) + bn .
It is well known that[2] |cn | ≤ 2 for all n ≥ 1 and |bn | ≤ 1 for all n ≥ 2. Therefore from (6), n|an | ≤ (1 − α) [|cn−1 | + |b2 | |cn−2 | + |b3 | |cn−3 | + · · · + |bn−1 | |c1 |] + |bn | = 2n(1 − α) + (2α − 1)
That is, |an | ≤ 2(1 − α) +
(2α − 1) for all n ≥ 2. n
Theorem 2. (Distortion Theorem.) 0 Let f ∈ C (α). Then (7)
1 − (1 − 2α)r 1 + (1 − 2α)r 0 ≤ |f (z)| ≤ , 2 (1 + r) (1 − r)2
|z| = r < 1.
A NOTE ON A SUBCLASS OF CLOSE-TO-CONVEX FUNCTIONS
65
0
Proof. Let f ∈ C (α). Then 0
zf (z) = (1 − α)p(z) + α, where p(0) = 1, Re p(z) > 0. g(z) (8)
0
zf (z) = [(1 − α)p(z) + α]g(z)
It is well known that [2, 3] |p(z)| ≤
1+r r ; and |g(z)| ≤ for |z| = r < 1. 1−r 1−r
Therefore from (8) 0
|zf (z)| ≤ [(1 − α)|p(z)| + α] |g(z)| r 1+r +α ≤ (1 − α) 1−r 1−r 1 + (1 − 2α)r = (1 − r)2 Similarly we can prove, 0
|f (z)| ≥
1 − (1 − 2α)r . (1 + r)2
Hence we have, 1 − (1 − 2α)r 1 + (1 − 2α)r 0 ≤ |f (z)| ≤ , 2 (1 + r) (1 − r)2
|z| = r < 1.
Theorem 3. (Growth Theorem) 0 Let f ∈ C (α). Then for |z| = r < 1, (9) 2(1 − α)r 2(1 − α)r −(1−2α)log(1+r) ≤ |f (z)| ≤ +(1−2α)log(1−r) (1 + r) (1 − r) 0
Proof. Let f ∈ C (α). Then 0
zf (z) = (1 − α)p(z) + α, where p(0) = 1, Re p(z) > 0. g(z)
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Now r Z 0 |f (z)| = f (z)dz 0
Zr ≤
0
|f (z)|dt 0
Zr ≤
1 + (1 − 2α)t dt from (7) (1 − t)2
0
2r(1 − α) + (1 − 2α)log(1 − r) (1 − r)
≤
Similarly we can prove that |f (z)| ≥
2r(1 − α) − (1 − 2α)log(1 + r) (1 + r)
Therefore, 2(1 − α)r 2(1 − α)r −(1−2α)log(1+r) ≤ |f (z)| ≤ +(1−2α)log(1−r), (1 + r) (1 − r) for |z| = r < 1.
Theorem 4. (Radius of Convexity) 0 Let f ∈ C (α). Then f is convex in |z| < ro where ro = 0 ≤ α < 1.
1 , 3−4α
0
To obtain radius of convexity for the class C (α), we require the following lemma of Libera [1] Lemma. [1] Let p(z) ∈ P, the class of functions with positive real part. Let η be any complex number with Re {η} ≥ 0 then for |z| = r, 0 ≤ r < 1. zp0 (z) 2r (10) p(z) + η ≤ (1 − r)[1 + r + Re{η}(1 − r)]
0
Proof. f ∈ C (α) implies 0
zf (z) = [(1 − α)p(z) + α] , p(0) = 1, Re p(z) > 0.
A NOTE ON A SUBCLASS OF CLOSE-TO-CONVEX FUNCTIONS
67
Logarithmic differentiation yields, 00
(11)
1+
0
0
zf (z) zp (z) zg (z) α = + where η = > 0. 0 f (z) [p(z) + η] g(z) 1−α
Therefore (12)
0 00 0 zf (z) zp (z) zg (z) Re 1 + 0 = Re + Re f (z) [p(z) + η] g(z)
Since g is convex in E, it is starlike of order 1/2 [5], therefore 0 zg (z) 1 (13) Re ≥ g(z) 1+r Application of Lemma [1] together with (13) in (12) gives 00 zf (z) 1 2r Re 1 + 0 ≥ − f (z) 1 + r (1 − r)[1 + r + Re{η}(1 − r)] where η =
α 1−α
> 0.
00 zf (z) (4α − 3)r2 + 2(2α − 1)r + 1 Re 1 + 0 = f (z) (1 − r)2 [1 − (2α − 1)r] > 0 if (4α − 3)r2 + 2(2α − 1)r + 1 > 0. Consider the equation (4α − 3)r2 + 2(2α − 1)r + 1 = 0. Then r=
−(2α − 1) ± 2(α − 1) (4α − 3)
Take −(2α − 1) + 2(α − 1) (4α − 3) 1 = − (4α − 3) 1 = , 0 ≤ α < 1. 3 − 4α
ro =
That is, 00
1 zf (z) } > 0 for |z| < ro where ro = , 0 ≤ α < 1. Re {1 + 0 f (z) 3 − 4α
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Putting α = 0 in the above four theorems we deduce the results of C.Selvaraj[4]. References [1] R.J. Libera, Some radius of convexity problems, Duke Math. J. , 31(1964), 143-153. [2] Z.Nehari, Conformal mapping McGraw Hill, publishing company, Newyork, 1 (1952). [3] M.S.Robertson, On the theory of univalent functions, Ann. Math. , 37(1936), 374-408. [4] C. Selvaraj, On a subclass of close-to-convex funcitons (to appear), South East Asian Bull. Math. , Springer-Verlag, 2001 / 2002. [5] Thomas H.Mac Gregor, The radius of close-to-convex functions of order 1/2 Proc.Amer.Math.soc., 14 (1963) 71 -76.
Department of Mathematics, Jaya Engineering College, Prakash Nagar, M.T.H Road, Avadi-, India, (
[email protected])
Department of Mathematics, L N Government College, Ponneri, 601 204, India
Department of Mathematics, Loyola College, Chennai 600 034
Proc. Symp. Geometric Function Theory, Dec 2002, pp. 69–74
ON THE WEIGHTED MEAN OF UNIVALENT CONVEX FUNCTIONS OF ORDER ρ. T. V. SUDHARSAN, HELEN C. DAVID, AND K. G. SUBRAMANIAN
Abstract. In this paper, functions hλ given by hλ (z) = 2−1 [(1 − λ)f (z) + (1 + λ)g(z)], 0 ≤ λ < 1 are considered, where f, g belong to the class K(ρ) of all normalized univalent analytic functions defined in the unit disc ∆ = {z/|z| = r < 1} and which are convex of order ρ, with 0 ≤ ρ < 1 and the radius of ρ - convexity of the functions is found.
1. Introduction Let S denote the class of all univalent analytic functions f defined in the unit disc ∆ : {z/|z| = r < 1} with f (0) = f 0 (0) − 1 = 0. For a fixed ρ, 0 ≤ ρ < 1, let K(ρ) denote the sub-class of S consisting of all functions f satisfying the condition zf 00 (z) Re 1 + 0 > ρ, |z| < 1. f (z) Functions in the class K(ρ) are known as univalent convex functions of order ρ [6]. Padmanabhan [4] has studied the problem of determination of the radius of ρ-convexity of the arithmetic mean of two functions in K(ρ) and proved that if f and g are two functions in K(ρ), then (f + g)/2 ∈ K(ρ), |z| < k, where k is the smallest positive root of the equation (1)
1 − 3r + 2r2 − 2r3 = 0.
Labelle and Rahman [1], have proved this result for ρ = 0. In fact, Padmanabhan [4] observes, quite interestingly, that this radius of convexity is independent of ρ and the result is not sharp. However, it is the aim of this paper to study the weighted mean hλ of functions in 69
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T. V. SUDHARSAN, HELEN C. DAVID, AND K. G. SUBRAMANIAN
the class K(ρ). More precisely we determine the radius of ρ-convexity for the weighted mean hλ (z) of f and g given by 1 hλ (z) = [(1 − λ)f (z) + (1 + λ)g(z)], 0 ≤ λ < 1, 2 where f, g ∈ K(ρ). The case ρ = 0, λ = 0 is Labelle and Rahman’s Theorem. It is to be noted that λ does not appear in the equation for the radius of ρ-convexity in the Theorem as given below, though ρ appears. The result is not sharp, as even in their case.
2. Main Result We need the following Lemmas in this connection: Lemma 1. ([3], Lemma 3). Let Pρ be the class of functions holomorphic in ∆ which satisfy P (0) = 1 and ReP (z) > ρ, 0 ≤ ρ < 1. Then 2 (1 + (1 − 2ρ)r ) P (z) − ≤ 2(1 − ρ)r , z ∈ ∆. (1 − r2 ) (1 − r2 ) In particular, we may take ([4], Lemma) P (z) = 1 +
zf 00 (z) , f ∈ K(ρ), z ∈ ∆. f 0 (z)
Lemma 2. ([2], p.111) If a < b, 0 < a, b < π/2, then sina/a > sinb/b. Theorem 1. Let f, g ∈ K(ρ) and hλ be defined as given above, then Re{1 + zh00λ (z)/h0λ (z)} > ρ, |z| < K, where K is the smallest positive root of the equation (2)
1 − 3r + 2(1 + 2ρ − ρ2 )r2 (1 − r) = 0.
Proof. With hλ (z) as in theorem and Cλ = (1 + λ)/(1 − λ), Rλ = Cλ R, we have, on writing g 0 (z)/f 0 (z) = Reiφ
z[(1 − λ)f 00 (z) + (1 + λ)g 00 (z)] (1 − λ)f 0 (z) + (1 + λ)g 0 (z) 00 f (z) g 00 (z)Rλ eiφ = z + (1 + Rλ eiφ )−1 f 0 (z) g 0 (z)
zh00λ (z)/h0λ (z) = (3)
WEIGHTED MEAN OF CONVEX FUNCTIONS
71
Writing (4)
ζ = (1 + Rλ eiφ )−1 = νeiψ
and with p(z) − 1 = w(z) = zf 00 (z)/f 0 (z), we find from Lemma 1,
(5)
|wζ − 2r2 (1 − ρ)ζ/(1 − r2 )| ≤ 2rν(1 − ρ)/(1 − r2 )
Thus, for |z| = r < 1, we have from (4) and (5)
Re[(zf 00 (z)/f 0 (z))(1 + Rλ eiψ )−1 ] = Re(wζ) (6) ≥ 2ν(1 − ρ)r(rcosψ − 1)/(1 − r2 ) where
(7)
ψ = arg(1 + Rλ eiφ )−1 = arc tan{Rλ sinφ/(1 + Rλ cosφ)},
(8)
ν = (1 + 2Rλ cosφ + Rλ2 )−1/2
so that (9)
cosψ = ν(1 + Rλ cosφ).
Again from Lemma 1, in a similar manner, on taking (10)
0
ν 0 eiψ = Rλ eiφ (1 + Rλ eiφ )−1
so that (11)
ν 0 = νRλ and cosψ 0 = ν(Rλ + cosφ),
Re[(zg 00 (z)/g 0 (z)){Rλ eiφ (1 + Rλ eiφ )−1 }] (12)
≥ {2r(1 − r2 )−1 }{(1 − ρ)ν 0 (rcosψ 0 − 1)} = 2(1 − ρ)νrRλ [(Rλ + cosφ)νr − 1](1 − r2 )−1 .
Using (6) and (12) in (3), we get
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T. V. SUDHARSAN, HELEN C. DAVID, AND K. G. SUBRAMANIAN
Re{zh00λ (z)/h0λ (z) + 1 − ρ} ≥ (1 − ρ) + 2νr(1 − ρ) [(r/ν) − 1 − Rλ ](1 − r2 )−1 = (1 − ρ) + (1 − ρ)[2r2 − 2νr(1 + Rλ )] (13) (1 − r2 )−1 . Again, if φ(z) ∈ K(ρ), |z| = r < 1, we have ([5], Theorem 5. p. 728) (14)
|arg φ0 (z)| ≤ 2(1 − ρ) arc sin r, arc sin 0 = 0,
which is a sharp estimate, for 0 ≤ ρ < 1, so that for f (z), g(z) ∈ K(ρ), |φ| = |arg(g 0 (z)/f 0 (z))| ≤ |arg g 0 (z)| + |arg f 0 (z)| ≤ 4(1 − ρ) arc sin r. √ Thus, for r < 1/ 2, cosφ ≥ cos[4(1 − ρ) arc sin r] (15) = 1 + 8sin4 [(1 − ρ) arc sin r] − 8sin2 [(1 − ρ) arc sin r]. Using Lemma 2, with a = (1 − ρ) arc sin r, b = arc sin r ≤ π/4. We get, since 0 ≤ ρ < 1, (16)
sin[(1 − ρ) arc sin r] ≥ (1 − ρ) sin(arc sin r) = (1 − ρ)r.
Therefore, (15) and (16) now give, since sint ≤ t (t ≥ 0), (17)
cosφ ≥ 1 + 8(1 − ρ)4 r4 − 8(1 − ρ) arc sin r)2 .
Now, on using (17), we have (18) 1+Rλ2 +2Rλ cosφ ≥ 1+Rλ2 +2Rλ {1+8(1−ρ)4 r4 −8(1−ρ)2 (arc sin r)2 } so that from (18), we obtain ν ≤ [1 + Rλ2 + 2Rλ {1 + 8(1 − ρ)4 r4 − 8(1 − ρ)(arc sin r)2 }]−1/2 (19) = Hλ (r), say.
WEIGHTED MEAN OF CONVEX FUNCTIONS
73
But 1
(1 + Rλ )Hλ (r) = [1 − 16Rλ (1 − ρ)2 (1 + Rλ )−2 {(arc sin r)2 − (1 − ρ)2 r4 }]− 2 (20)
1
≤ [1 − 4(1 − ρ)2 {(arc sin r)2 − (1 − ρ)2 r4 }]− 2 ,
Finally, from (13), (19) and (20) we get
Re[1 + zhλ (z)/h0λ (z) − ρ] ≥ (1 − ρ)[(1 + r2 ) − 2r(1 − 4(1 − ρ)2 (arc sin r)2 + 4(1 − ρ)4 r4 )−1/2 ] ≥ 0, if (21)
(1 + r2 )2 [1 − 4(1 − ρ)2 (arc sin r)2 + 4(1 − ρ)4 r4 ] ≥ 4r2
Since arc sin r ≥ r ≥ 0, we have from (21) (1 + r2 )2 [1 − 4(1 − ρ)2 r2 + 4(1 − ρ)4 r4 ] ≥ 4r2 , or (1 + r2 )[1 − 2(1 − ρ)2 r2 ] ≥ 2r. Thus 1 − 2r + (1 − 2(1 − ρ)2 )r2 − 2(1 − ρ)2 r4 ≥ 0 or 1 − 2r + (1 − 2(1 − ρ)2 )r2 + (2(1 − ρ)2 − 4)r4 ≥ 0, since 2(1 − ρ)2 − 4 ≥ −2(1 − ρ)2 , 0 ≤ ρ < 1, or (1 + r)(1 − 3r + 2(2 − (1 − ρ)2 )r2 + (2(1 − ρ)2 − 4)r3 ) ≥ 0 √ so that from (22) with r < 1/ 2 < 1, the radius of convexity is the smallest positive root K of the equation (22)
1 − 3r + 2(1 + 2ρ − ρ2 )r2 (1 − r) = 0, which is (2) of Theorem. Remark 1. The above theorem does not however provide a sharp result but provides a more interesting result when compared to that of Padmanabhan [4] which is just the case λ = 0, but without ρ in (2) of Theorem 1.
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T. V. SUDHARSAN, HELEN C. DAVID, AND K. G. SUBRAMANIAN
Acknowledgement: The work was started with the guidance and support of Dr. T.V. Lakshminarasimhan, former professor of Mathematics, Madras Christian College, Chennai 600 059 for which the authors are very grateful although he is no more with us to see the final version. References [1] G. Labelle and Q.I. Rahman, Remarque sur la moyenne arithmetique de functions univalentes conexes, Canad J Mathematics 21 (1969), 978-981. [2] Mitrinovic, Elementary inequalities, P Noordhoff Ltd. - Groningen, The Netherlands (1964). [3] Michael R Ziegler, Some integrals of univalent functions, Indian J Mathematics, 11 (1969), 145-151. [4] K.S. Padmanabhan, On the arithmetic mean of univalent convex functions, 65-68 Glasnik Mathematicki 9 (29) (1974). [5] Pinchuk, On starlike and convex functions of order α, Duke Math. Journal, 35 (1968), 721-734. [6] M.S. Robertson, On the theory of Univalent Functions, Ann. Math. (2) 37 (1936), 374-408.
Department of Mathematics, SIVET College, Gowrivakkam, Chennai 601 302
Department of Mathematics, Alpha Arts and Science College, Chennai 600 116
Department of Mathematics, Madras Christian College, Chennai 600 059
Proc. Symp. Geometric Function Theory, Dec 2002, pp. 75–84
AN APPLICATION OF FRACTIONAL CALCULUS OPERATORS TO A SUBCLASS OF p-VALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS S.R.KULKARNI AND SAYALI S.JOSHI
Abstract. The systematic investigation of aP new class Tp (α, β, λ, µ, k) ∞ consisting of functions of the form f (z) = z p − n=1 ap+n z p+n ), ap+n ≥ 0, p ∈ N which are p-valent in unit disc U and satisfying the condition f (z) − 1 Ωk,p z <β k,p (2λΩk,p z f (z) − α(1 + µ)λ) − (Ωz f (z) − 1) where 0 < λ ≤ 1, 0 ≤ µ ≤ 1, 0 ≤ α < 1, 0 < β ≤ 1, and Γ(1 + p − k) k−p k Ωk,p z Dz f (z), z f (z) = Γ(1 + p) where (Dzk is the fractional derivative operator of order k) is presented. Apart from various coefficient bounds and number of characterization and distortion theorems, many interesting and useful properties of this class of functions are obtained. 2000 Mathematics Subject Classification 30C45 Key words: Fractional calculus, coefficient bounds, distortion theorem
1. Introduction and Definitions Let Sp denote the class of functions of the from (1)
p
f (z) = z +
∞ X
ap+n z p+n (p ∈ N = {1, 2, ...})
n=1
which are analytic and p-valent in the open unit disk U = {z : |z| < 1}. Also let Tp denote the subclass of Sp consisting of analytic and p-valent 75
76
S.R.KULKARNI AND SAYALI S.JOSHI
function, which can be expressed, in the form (2)
p
f (z) = z −
∞ X
ap+n z p+n
(ap+n ≥ 0, p ∈ N )
n=1
The object of the present paper is to investigate a new class Tp (α, β, λ, µ, k) of analytic and p-valent functions f (z) belonging to the class Tp and satisfying condition Ωk,p z f (z) − 1 (3) (2λΩk,p f (z) − α(1 + µ)λ) − (Ωk,p f (z) − 1) < β z z (z ∈ U, 0 < λ ≤ 1, 0 ≤ µ ≤ 1, 0 ≤ α < 1, 0 < β ≤ 1) where for convenience, (4)
Ωk,p z f (z) =
Γ(1 + p − k) k−p k z Dz f (z), Γ(1 + p)
in terms of the fractional derivative operator Dzk of order k studied by Owa[3] and others (Srivastava, Owa [5]) defined below with Dz0 f (z) = f (z) and Dz1 f (z) = f 0 (z). Definition 1. (Fractional Integral Operator). The fractional integral of order k is defined, for a function f (z), by Z z 1 f (ξ) −k (5) Dz f (z) = dξ (k > 0) Γk 0 (z − ξ)1−k where f (z) is an analytic function in a simply connected region of the zplane containing the origin and the multiplicity of (z −ξ)k−1 is removed by requiring log(z − ξ) to be real when z − ξ > 0. Definition 2. (Fractional Derivative Operator) The fractional derivative of order k is defined, for a function f (z), by Z z d f (ξ) 1 k (6) Dz f (z) = dξ (0 ≤ k < 1) Γ(1 − k) dz 0 (z − ξ)k Definition 3. (Extended Fractional Derivative Operator). Under the hypothesis of Definition 2, the fractional derivative of order n + k is defined, for a function f (z), by dn k D f (z) (0 ≤ k < 1; n ∈ N0 = N ∪ {0}) dz n z where, N denotes the set of natural numbers. (7)
Dzn+k f (z) =
p-VALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS
77
For the function belonging to the general class Tp (α, β, λ, µ, k) we prove number of sharp results including for example coefficient estimate, distortion theorem, Closure theorem, and radius of starlikeness. By specializing parameters we get results earlier obtained by Ahmad and Sharma [1], Srivastava and Aouf [4].
2. Coefficient Estimate Theorem 1. A function f (z) defined by (2) is in class Tp (α, β, λ, µ, k) if and only if (8) ∞ X Γ(n + 1 + p)Γ(1 + p − k) (1 + β(2λ − 1))ap+n ≤ βλ(2 − α(1 + µ)). Γ(1 + p)Γ(n + p + 1 − k) n=1 The condition (8) is sharp.
Proof. Assume that the inequality (8) holds true and Let |z| = 1. Then we obtain k,p |Ωk,p f (z) − 1| − β|(2λΩk,p z f (z) − α(1 + µ)λ) − (Ωz f (z) − 1)| z∞ X Γ(p + n + 1)Γ(1 + p − k) = − ap+n z n Γ(1 + p)Γ(n + p + 1 − k) n=1 " # ∞ X Γ(n + 1 + p)Γ(1 + p − k) ap+n z n ] − α(1 + µ)λ −β 2λ[1 − Γ(1 + p)Γ(n + p + 1 − k) n=1 ∞ X Γ(p + n + 1)Γ(1 + p − k) −[− ap+n z n ] Γ(1 + p)Γ(n + p + 1 − k) n=1
≤
∞ X Γ(p + n + 1)Γ(1 + p − k) n=1
Γ(1 + p)Γ(n + p + 1 − k)
(1 + β(2λ − 1))ap+n − β[2λ − α(1 + µ)λ]
≤ 0 by hypothesis. Hence, by maximum modulus theorem we have f (z) ∈ Tp (α, β, λ, µ, k).
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S.R.KULKARNI AND SAYALI S.JOSHI
To prove the converse, assume that f (z) is defined by (2) and is in the class Tp (α, β, λ, µ, k), so that the condition (4) readily yields. k,p Ω f (z) − 1 z (9) 2(λΩk,p f (z) − α(1 + µ)λ) − Ωk,p f (z) − 1 z z ∞ X Γ(p + n + 1)Γ(1 + p − k) = ap+n z n × n=1 Γ(1 + p)Γ(n + p + 1 − k) −1 ∞ X Γ(n + 1 + p)Γ(1 + p − k) n ap+n z < β (2λ − α(1 + µ)λ) − (2λ − 1) Γ(1 + p)Γ(n + p + 1 − k) n=1 Since |Re(z)| ≤ |z|, we find from (9) that (" ∞ # X Γ(p + n + 1)Γ(1 + p − k) (10) Re ap+n z n Γ(1 + p)Γ(n + p + 1 − k) n=1 #−1 " ∞ X Γ(n + 1 + p)Γ(1 + p − k) n ap+n z < β. (2λ − α(1 + µ)λ) − (2λ − 1) Γ(1 + p)Γ(n + p + 1 − k) n=1 Choose value of z on the real axis so that (Ωk,p z f (z) is real. Upon clearing the denominator in (10) and letting z → 1 through real values, we have ∞ X Γ(p + n + 1)Γ(1 + p − k) (11) ap+n Γ(1 + p)Γ(n + p + 1 − k) n=1 ≤ βλ(2 − α(1 + µ)) − β(2λ − 1)
∞ X Γ(n + 1 + p)Γ(1 + p − k) n=1
Γ(1 + p)Γ(n + p + 1 − k)
ap+n
which gives desired assertion (8). Finally we note that the assertion (8) of the Theorem 1 is sharp, the extremal function being,
(12)
f (z) = z p −
βλ(2 − α(1 + µ))Γ(1 + p)Γ(n + p + 1 − k) n+p z (1 + β(2λ − 1))Γ(1 + p − k)Γ(1 + p + n)
Corollary 1. Let the function f (z) be defined by (2) be in the class Tp (α, β, λ, µ, k), then (13)
an+p ≤
βλ(2 − α(1 + µ))Γ(1 + p)Γ(n + p + 1 − k) (1 + β(2λ − 1))Γ(1 + p − k)Γ(1 + p + n)
p-VALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS
79
for every integer n ∈ N .
3. Distortion Theorem Theorem 2. Let the function f (z) be defined by (2) be in the class Tp (α, β, λ, µ, k) then (14)
|f (z)| ≥ |z|p − |z|p+1
(1 + p − k)βλ(2 − α(1 + µ)) , (1 + p)[1 + β(2λ − 1)]
(15)
|f (z)| ≤ |z|p + |z|p+1
(1 + p − k)βλ(2 − α(1 + µ)) , (1 + p)[1 + β(2λ − 1)]
for z ∈ U . Furthermore (16) |Dzk f (z)| ≥
Γ(1 + p) βλ[2 − α(1 + µ)]Γ(1 + p − k) |z|p−k − Γ(1 + p − k) [1 + β(2λ − 1)]Γ(1 + p)
(17) |Dzk f (z)| ≤
βλ[2 − α(1 + µ)]Γ(1 + p − k) Γ(1 + p) |z|p−k + Γ(1 + p − k) [1 + β(2λ − 1)]Γ(1 + p)
whenever z ∈ U . Proof. Since f (z) ∈ Tp (α, β, λ, µ, k) in view of Theorem 1, we have (18) ∞ X Γ(n + 1 + p)Γ(1 + p − k) (1 + p)[1 + β(2λ − 1)] X ap+n ≤ [1+β(2λ−1)]ap+n 1+p−k Γ(p + 1)Γ(n + p + 1 − k) n=1 ≤ βλ[2 − α(1 + µ)], which yields in (19)
∞ X n=1
ap+n ≤
βλ[2 − α(1 + µ)](1 + p − k) . (1 + p)[1 + β(2λ − 1)]
Consequently, we obtain (20) ∞ X βλ[2 − α(1 + µ)](1 + p − k) p+1 p p+1 |f (z)| ≥ |z| −|z| ap+n ≥ |z|p − |z| . (1 + p)[1 + β(2λ − 1)] n=1
80
S.R.KULKARNI AND SAYALI S.JOSHI
and (21) |f (z)| ≤ |z|p +|z|p+1
∞ X
ap+n ≤ |z|p +
n=1
βλ[2 − α(1 + µ)](1 + p − k) p+1 |z| . (1 + p)[1 + β(2λ − 1)]
which proves the assertion (14) and (15) of the Theorem 2. Now, by using second inequality on (18), we observe that
(22)|z
p
Ωk,p z f (z)|
p
≥ |z| −
∞ X Γ(n + 1 + p)Γ(1 + p − k) n=1
p
Γ(p + 1)Γ(n + p + 1 − k)
p+1
≥ |z| − |z|
∞ X Γ(n + 1 + p)Γ(1 + p − k) n=1
≥ |z|p − |z|p+1
ap+n |z|n+p
Γ(p + 1)Γ(n + p + 1 − k)
ap+n
βλ[2 − α(1 + µ)] , [1 + β(2λ − 1)]
and (23)|z
p
Ωk,p z f (z)|
p
≤ |z| +
∞ X Γ(n + 1 + p)Γ(1 + p − k) n=1
p
Γ(p + 1)Γ(n + p + 1 − k)
p+1
≤ |z| + |z|
∞ X Γ(n + 1 + p)Γ(1 + p − k) n=1
≤ |z|p + |z|p+1
ap+n |z|n+p
Γ(p + 1)Γ(n + p + 1 − k)
ap+n
βλ[2 − α(1 + µ)] , [1 + β(2λ − 1)]
which prove the assertion (16) and (17) of Theorem 2.
4. Closure Theorems We begin by defining the function, fj (z)(j = 1...m) by (24)
p
fj (z) := z −
∞ X n=1
ap+n,j z p+n (ap+n,j ≥ 0, p ∈ N, z ∈ U ).
p-VALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS
81
Theorem 3. Let the function fj (z)(j = 1, ...m) defined by (24) be in the classes Tp (αj , βj , λj , µj , k)(j = 1, ...m), respectively. Then the function h(z) defined by ! ∞ m X X 1 (25) h(z) = z p − ap+n,j z p+n m n=1 j=1 is in the class Tp (α, β, λ, µ, k) where (26)
α = min1≤j≤m (αj ), λ = max1≤j≤m (λj ),
β = max1≤j≤m (βj ), µ = min1≤j≤m (µj )
Proof. Since fj ∈ Tp (αj , βj , λj , µj , k)(j = 1, ...m) , then by using Theorem 1, to definition (24), we get ∞ m X Γ(n + 1 + p)Γ(1 + p − k) 1 X [ ap+n,j ] Γ(p + 1)Γ(n + p + 1 − k) m n=1 j=1 m
∞
=
1 X X Γ(n + 1 + p)Γ(1 + p − k) ap+n,j m j=1 n=1 Γ(p + 1)Γ(n + p + 1 − k)
≤
1 X βj λj [2 − αj (1 + µj )] , m j=1 [1 + βj (2λj − 1)]
≤
βλ[2 − α(1 + µ)] [1 + β(2λ − 1)]
m
where α, β, λ, and µ are given by (26). Thus we have (27) ∞ m X 1 X Γ(n + 1 + p)Γ(1 + p − k) [1+β(2λ−1)][ ap+n,j ] ≤ βλ[2−α(1+µ)] Γ(p + 1)Γ(n + p + 1 − k) m j=1 n=1 which in view of Theorem 1, implies that h(z) ∈ Tp (α, β, λ, µ, k). The proof of Theorem 3 is completed. Next we state and prove Theorem 4. The class Tp (α, β, λ, µ, k) is convex. Proof. Let the function fj (z)(j = 1, 2) defined by (24) are in the class Tp (α, β, λ, µ, k) Then it is sufficient to prove that (28)
h(z) = ηf1 (z) + (1 − η)f2 (z)(0 ≤ η ≤ 1)
82
S.R.KULKARNI AND SAYALI S.JOSHI
or equivalently p
h(z) = z −
∞ X
{ηap+n,1 + (1 − η)ap+n,2 }z p+n
(0 ≤ η ≤ 1)
n=1
is also in the class Tp (α, β, λ, µ, k). Now, from our hypothesis and Theorem 1, it follows readily that ∞ X n=1
Γ(n + 1 + p)Γ(1 + p − k) [1+β(2λ−1)]{ηap+n,1 +(1−η)ap+n,2 } ≤ βλ[2−α(1+µ)] Γ(p + 1)Γ(n + p + 1 − k)
which evidently proves Theorem 4. Theorem 5. Let fp (z) = z p , p ∈ N
(29) and (30) fp+n (z) = z p −
βλ[2 − α(1 + µ)]Γ(p + 1)Γ(n + p + 1 − k) p+n z [1 + β(2λ − 1)]Γ(n + 1 + p)Γ(1 + p − k)
then the function f (z) is in the class Tp (α, β, λ, µ, k) if and only if it can be expressed in the form (31)
f (z) =
∞ X
Cp+n fp+n ,
Cp+n ≥ 0,
n=1
∞ X
Cp+n = 1
n=0
. Proof. First of all, let us assume that f (z) given by (31), that is by (32) ∞ X βλ[2 − α(1 + µ)]Γ(p + 1)Γ(n + p + 1 − k) f (z) = z p − Cp+n z p+n [1 + β(2λ − 1)]Γ(n + 1 + p)Γ(1 + p − k) n=1
(Cp+n ≥ 0, .
∞ X
Cp+n = 1)
n=0
Then, since (33) ∞ X Γ(n + p + 1)Γ(p + k − 1)[1 + β(2λ − 1)]βλ[2 − α(1 + µ)]Γ(p + 1)Γ(n + p + 1 − k) Cp+n Γ(1 + p)Γ(p + n + 1 − k)[1 + β(2λ − 1)]Γ(n + 1 + p)Γ(1 + p − k) n=1
p-VALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS
= βλ[2−α(1+µ)]
∞ X
83
Cp+n = βλ[2−α(1+µ)](1−Cp ) ≤ βλ[2−α(1+µ)],
n=1
Theorem 1, implies that f ∈ Tp (α, β, λ, µ; k). Conversely assume that the function f (z) defined by (2) is in the class Tp (α, β, λ, µ; k) then, using (13) of corollary 1, we have (34)
ap+n ≤
βλ[2 − α(1 + µ)]Γ(p + 1)Γ(n + p + 1 − k) [1 + β(2λ − 1)]Γ(n + 1 + p)Γ(1 + p − k)
and setting (35)
Cp+n =
[1 + β(2λ − 1)]Γ(n + 1 + p)Γ(1 + p − k) ap+n βλ[2 − α(1 + µ)]Γ(p + 1)Γ(n + p + 1 − k)
and (36)
Cp = 1 −
∞ X
Cp+n ,
n=1
we can readily see that f (z) can be expressed precisely as in (32). This evidently completes the proof of Theorem 5. Theorem 6. Let the function f (z) defined by (2) be in the class Tp (α, β, λ, µ; k). Then f (z) is p-valent starlike of order δ(0 ≤ δ < p)in the disc |z| < r1 , where (37) 1/n (p − δ)[1 + β(2λ − 1)]Γ(1 + p − k)Γ(1 + p + n) r1 = r1 (p, α, β, µ, λ, k, δ) = inf . (n + p − δ)βλ[2 − α(1 + µ)]Γ(p + 1)Γn + p + 1 − k The result (37) is sharp, with extreme function given by (12). Proof. Making use of the definition (2), we readily observe that P∞ n=1 nap+n|z|n 0 0 P ≤ (p − δ). (38) |zf (z)/f (z) − p| ≤ n 1− ∞ n=1 ap+n |z| If ∞ X n+p−δ [ ]ap+n |z|n ≤ 1 p − δ n=1
84
S.R.KULKARNI AND SAYALI S.JOSHI
. that is if, (39) n+p−δ n [1 + β(2λ − 1)]Γ(1 + p − k)Γ(1 + p + n) [ ]|z| ≤ , n∈N p−δ βλ[2 − α(1 + µ)]Γ(p + 1)Γ(n + p + 1 − k) ,where we have also applied the assertion (8) of Theorem 1. The last inequality (39) lead us precisely to the disc |z| < r, where r, is given by (37), and the proof of Theorem 6 is evidently completed. References [1] Iqbal Ahmad and Y.K.Sharma, p-valent function with negative coefficients. The Math Student, 63, 1-4(1994), 237-242. [2] Goel R.M. and Sohi N.S. Multivalent function with negative coefficients. Indian J. Pure. App. Math 12(7), (1981), 844-854. [3] S.Owa, On the distortion Theorem I, Kyungpook Math J. 18 (1978), 53-59. [4] H.M.Srivastava and M.K.Aouf, A Certain Fractional derivative operator and its application to a new class of analytic and multivalent function with negative coefficients, J.Math.Anal.Appl, 171 (1992), 1-13. [5] H.M.Srivastava and S.Owa, A new class of analytic function with negative coefficients. Comment. Math. Univ. St. Paul 35(1986), 175-188. [6] H.M.Sivastava and S.Owa (Editor)Univalent Functions, Fractional Calculus of their Applications Halsted Press (Ellis Harwood United, Chichester), Wiley, New York/Chicherster/Brisbane/Toronto 1989
Department of Mathematics, Fergusson College, Pune 411 004
F-7, Shivam Aptts, Warnali Road, Vishrmbag, Sangli 416 415 E-mail address:
[email protected]
Proc. Symp. Geometric Function Theory, Dec 2002, pp. 85–94
A DIFFERENTIAL OPERATOR AND ITS APPLICATIONS TO A GENERAL AND NOVEL CLASS OF MEROMORPHICALLY UNIVALENT FUNCTIONS WITH NEGATIVE OR POSITIVE COEFFICIENTS S B JOSHI
Abstract. Using a differential operator, we introduce a class (A,B) (q; α) of meromorphically univalent functions with negaMp tive or positive coefficients in the punctured unit disk D, and obtain a necessary and sufficient condition for functions in the class (A,B) (q; α) include the meromorphically starlikeness, meromprphiMp cally convexity, and some growth and distortion theorems involving generalized fractional operators. 2000 AMS Subject Classification: 30C45 Key words and phrases: Meromorphic functions differential operator, coefficient bounds, growth and distortion theorems, punctured unit disc, maximum modulus theorem, and radii of meromorphically starlikeness and meromorphically convexity.
1. Introduction and Definitions (A,B)
Let Mp (1) f (z) =
denote the class of functions f (z) of the form:
∞ X A +B an z n , z n=p
(an ≥ 0; p ∈ N = {1, 2, 3, . . .}; A.B 6= 0),
which are analytic and univalent in the punctured unit disk D = {z; z ∈ C and 0 < |z| < 1}, and which have a simple pole at the origin (z = 0) 85
86
S B JOSHI (A,B)
with residue A there. A function f (z) ∈ Mp is said to be meromorphically starlike of order α in D if it satisfies the inequality: 0 zf (z) (2) −Re > α, (0 ≤ α < 1; z ∈ D). f (z) (A,B)
Furthermore, a function f (z) ∈ Mp is said to be meromorphically convex of order α in D if it satisfies the inequality: zf 00 (z) (3) −Re 1 + 0 > α, (0 ≤ α < 1; z ∈ D). f (z) A function f (z) is said to be meromorphically convex of order α (0 ≤ α < 1) if and only if zf 0 (z) is meromorphically starlike of order α (0 ≤ α < 1). ( See, for details, Duren [7], Goodman [8]; see (also) Nehari [9] and Pommerenke [10].) (A,B)
(A,B)
A function f (z) ∈ Mp is said to be in the class Mp also satisfy the inequality: zf (q+1) (z) < α, (4) − − (q + 1) f (q) (z)
(q; α) if it
(0 < α ≤ 1; z ∈ D; p ≥ q; p ∈ N; q ∈ N0 = N ∪ {0}). where (5) ∞ X n! Aq!(−1)q (q) f (z) = +B an z n−q , (p ≥ q; p ∈ N, q ∈ N0 ). q+1 z (n − q)! n=p (A,B)
In view of the above definition of the class Mp (q; α), it is easily (A,B) verified that the class Mp (q; α) can be identified with the class of (i) Meromorphically starlike functions of order 1 − α (0 < α ≤ 1) when q = 0. (ii) Meromorphically convex functions of order 1 − α (0 < α ≤ 1) when q = 1. (A,B)
Some other interesting subclasses of the class Mp were studied recently by Chen et al. [2,3], Altinates et al. [1], Cho et al [5,6], and others [4], [2] and [14].
A DIFFERNTIAL OPERATOR
87
2. Coefficient Bounds for Functions in the class (A,B) Mp (q; α) (A,B)
A necessary and sufficient condition for function f (z) ∈ Mp with A = B = 1 is provided by
(q; α)
(1,1)
Theorem 1. Let a function f (z) be in the class Mp . Then, the (1,1) function f (z) belongs to the class Mp (q; α) if and only if (6) ∞ X n![n + 1 − (−1)−q α] an ≤ αq!, (p ≥ q; p ∈ N, q ∈ N0 , α ≤ 1). (n − q)! n=p The result is sharp, the extremal function being given by (7)
f (z) =
1 αq!(p − q)! + zp, z p![p + 1 − (−1)−q α]
(p ≥ q; p ∈ N, q ∈ N0 ).
(1,1)
Proof. Suppose that the function f (z) ∈ Mp , and the inequality (6) holds true. Then, making use of the differential operator defined by (5) with A = B = 1, we find for z ∈ ∂D that |zf (q+1) (z) + (q + 1)f (q) (z)| − α|f (q) (z)| ( ) ∞ ∞ X X n! n! an z n−q − α (−1)q q! + (−1)−q an z n−q = (n − q − 1)! (n − q)! ≤
n=p ∞ X n=p
n=p
−q
n![n + 1 − (−1) α] an − αq! ≤ 0, (n − q)!
by virtue of the inequality (6). Hence, by the maximum modulus the(1,1) orem, we have f (z) ∈ Mp (q, α). To prove the converse, we suppose that f (z) is defined by (1) with (1,1) A = B = 1, and is in the class Mp (q; α), so that the condition (4) readily yields: (8) P∞ (q+1) n! n+1 (q) zf (z) + (q + 1)f (z) n=p (n−q−1)! an z = P < α, ∞ n! (q) −q n+1 q! + (−1) f (z) n=p (n−q)! an z (z ∈ D; p ≥ q; p ∈ N, q ∈ N0 , 0 < α ≤ 1).
88
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Since |Re(z)| ≤ |z| for any z, if we choose z to be real and z → 1−, we shall find from (8) that ( ) ∞ ∞ X X n! n! an ≤ α q! + (−1)−1 an , (n − q − 1)! (n − q)! n=p n=p which readily yields the desired assertion (6) of Theorem 1. Finally, by observing that the function f (z) given by (7), is indeed extremal function for the assertion (6), we complete the proof of Theorem 1. (1,−1)
Theorem 2. Let a function f (z) be in the class Mp . Then, the (1,−1) function f (z) belongs to the class Mp (q; α) if and only if ∞ X n! (9) an ≤ αq!, (p ≥ q; p ∈ N, q ∈ N0 , 0 < α ≤ 1). (n − q)! n=p The result is sharp, the extremal function being given by 1 αq!(p − q)! (10) f (z) = − z p , (p ≥ q; p ∈ N, q ∈ N0 ). z p![p + 1 + (−1)−q α] The following corollaries are rather immediate consequences of Theorem 1 and Theorem 2, respectively. (1,1)
Corollary 1. If f (z) ∈ Mp (11)
an ≤
(q, α), then
αq!(n − q)! , p![p + 1 + (−1)−q α] (1,−1)
Corollary 2. If f (z) ∈ Mp (12)
an ≤
(n > p ≥ q; p ∈ N, q ∈ N0 ).
(q, α), then
αq!(n − q)! , n![n + 1 + (−1)−q α]
(n > p ≥ q; p ∈ N, q ∈ N0 ).
3. Growth and Distortion Theorems for Functions (1,1) belonging to the Class Mp (q, α) In this section, we shall prove several growth and distortion theo(1,1) rems for functions belonging to the calss Mp (q, α). Each of these theorems would involve generalized fractional calculus (that is generalized fractional integral and generalized fractional derivative), which are defined as follows (c.f., e.g., [11], and see (also) [13]).
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Definition 1. Let β ∈ R+ and φ, η ∈ R. Then, in terms of the familiar (Gauss’s) hypergeometric function 2 F1 , the generalized fractional β,φ,η is defined by integral operator I0,z Z z −β−φ z t β,φ,η (13) I0,z {g(z)} = (z − t)β−1 g(t)2 F1 (β + φ, −η; β; 1 − )dt, Γ(β) 0 z where the function g(z) is an analytic in simply-connected region of the z-plane containing the origin, with the order (14)
g(z) = O(|z| ),
(z → 0),
for > max{0, φ − η} − 1, and the multiplicity of (z − t)β−1 is removed by requiring log(z − t) to be real when z − t > 0. Definition 2. Let 0 ≤ β < 1 and φ, η ∈ R. Then, the generalized β,φ,η is defined for a function g(z), by fractional derivative operator J0,z β,φ,η J0,z {g(z)}
(15) ( =
R d β−φ z 1 z (z − t)β−1 g(t)2 F1 (β Γ(1−β) dz 0 ν d J β,φ,η {g(z)} (ν ≤ β < ν + 1; ν dz ν 0,z
+ φ, −η; β; 1 − zt )dt, ∈N
(0 ≤ β < 1)
where the function g(z) is analytic in a simply-connected region of the z-plane containing the origin, with the order as given by (14), and the multiplicity of (z − t)−β is removed by the requiring log(z − t) to be real when z − t > 0. From the definition 1 and definition 2 it is clear that (c.f., [12,13]) : (16) (17)
β,−β,η I0,z {g(z)} = Dz−β {g(z)}, β,β,η J0,z {g(z)}
= Dzβ {g(z)},
(β > 0); (0 ≤ β).
Now we state two auxiliary results. Lemma 1. ([14], p. 415, Lemma 3). If µ > 0 and m > β − η − 1, then Γ(m + 1)Γ(m − β + η + 1) µ,β,η {z m } = (18) I0,z z m−β . Γ(m − β + 1)Γ(m − µ + η + 1) Lemma 2. ([12], p. 15, Eq. (2.2), or, [4], p. 85, Eq. (2.3)). If 0 ≤ µ < 1 and m > β − η − 1, then Γ(m + 1)Γ(m − β + η + 1) µ,β,η (19) J0,z {z m } = z m−β . Γ(m − β + 1)Γ(m − µ + η + 1)
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Now, we can give and prove some growth and distortion theorems β,φ,η β,φ,η involving the generalized fractional operators I0,z and J0,z respectively. Theorem 3. Let µ > 0 and β, η ∈ R such that β < 2, µ + η + 2 > 0, β − η < 2. If n is positive integer such that n≥
(20) (1,1)
and, if f (z) ∈ Mp (21)
β(µ + η) − 2, µ (1,1)
is in the class Mp
(q; α), then
µ,β,η I0,z {z q+2 f (q) (z)} q!Γ(2 − β + η) − |z|1−β Γ(2 − β)Γ(2 + µ + η) αq!(p + 2)!Γ(p − β + η + 3) ≤ |z|p−β+2 Γ(p − β + η + 3)Γ(p + µ + η + 3)[p + 1 − (−1)q α]
for z ∈ U if β ≤ 1 and z ∈ U − {0} if β > 1. The inequalities in (21) are obtained by the function f (z) given by (7) . (1,1)
Proof. Suppose that f (z) ∈ Mp (q, α). Under the hypotheses of Theorem 2, it follows from the inequality (6) that [p + 1 − (−1)q α]
∞ X n=p
∞
X n![n + 1 − (−1)q α] n! an ≤ an ≤ αq! (n − q)! (n − q)! n=p
which evidently yields (22)
∞ X n=p
αq! n! an ≤ , (n − q)! P + 1 − (−1)−q α
(p > q; p ∈ N; q ∈ N0 ).
From which (1) and Lemma 1, we have that µ,β,η I0,z {z q+2 f (q) (z)} =
(23)
Γ(2 − β + η) z 1−β Γ(2 − β)Γ(2 + µ + η) ∞ X n!Γ(n + 3)Γ(n − β + η + 3)
an z n−β+2 (n − q)!Γ(n − β + η + 3)Γ(n + µ + η + 3) n=p ( ) ∞ X Γ(2 − β + η) n! 1−β n+1 = z − Θ(n) an z Γ(2 − β)Γ(2 + µ + η) n=p (n − q)!
+
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where, for convenience, Θ(n) =
Γ(n + 3)Γ(n − β + η + 3) , Γ(n − β + η + 3)Γ(n + µ + η + 3)
(n = p, p + 1, . . . ; p ∈ N).
Clearly, the function Θ(n) is decreasing for n and we have that (24)
0 < Θ(n) ≤ Θ(p) =
Γ(p + 3)Γ(p − β + η + 3) , Γ(p − β + 3)Γ(p + µ + η + 3)
p ∈ N.
Thus from (22) - (24), it is easily seen that Γ(2 − β + η) µ,β,η q+2 (q) z 1−β I0,z {z f (z)} = Γ(2 − β)Γ(2 + µ + η) ∞ X n!Γ(n + 3)Γ(n − β + η + 3) n−β+2 + an z (n − q)!Γ(n − β + η + 3)Γ(n + µ + η + 3) n=p ∞ X n! Γ(2 − β + η) = |z|1−β Θ(n) + an |z|n+1 Γ(2 − β)Γ(2 + µ + η) (n − q)! n=p ∞ X n! Γ(2 − β + η) p+1 1−β ≥ |z| + Θ(p)|z| an Γ(2 − β)Γ(2 + µ + η) (n − q)! n=p Γ(2 − β + η) ≥ |z|1−β Γ(2 − β)Γ(2 + µ + η) αq!(p + 2)Γ(p − β + η + 3) p−β+1 − |z| Γ(p − β + 3)Γ(p + µ + η + 3)[p + 1 − (−1)q α]
which completes the proof of Theorem 3. (1,−1)
Theorem 4. Under the hypothesis of Theorem 3, if f (z) ∈ Mp is (1,−1) in the class Mp (q, α), then q!Γ(2 − β + η) µ,β,η q+2 (q) 1−β (25) |z| |I0,z {z f (z)}| − Γ(2 − β)Γ(2 + µ + η) αq!(p + 2)!Γ(p − β + η + 3) ≤ |z|p−β+2 Γ(p − β + η + 3)Γ(p + µ + η + 3)[p + 1 − (−1)q α] for z ∈ U if β ≤ 1 and z ∈ U − {0} if β > 1. The inequalities in (25) are obtained by the function f (z) given by (10) . We can prove Theorem 5 and 6 below in a similar way to those of Theorem 4 by making use of Lemma 2.
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Theorem 5. Let 0 ≤ µ < 1 and β, η ∈ R such that β < 2, µ − η < 2, and β − η < 2. If n is positive integer such that n≥
(26)
β(µ − η) − 2, µ
(1,1)
(1,1)
and, if f (z) ∈ Mp is in the class Mp (q; α), then q!Γ(2 − β + η) µ,β,η q+2 (q) 1−β |z| |J0,z {z f (z)}| − Γ(2 − β)Γ(2 + µ + η) αq!(p + 2)!Γ(p − β + η + 3) ≤ (27) |z|p−β+2 Γ(p − β + 3)Γ(p − µ + η + 3)[p + 1 − (−1)q α] for z ∈ U if β ≤ 1 and z ∈ U − {0} if β > 1. The inequalities in (27) are obtained by the function f (z) given by (7) . (1,−1)
Theorem 6. Under the hypothesis of Theorem 3, if f (z) ∈ Mp is (1,−1) in the class Mp (q, α), then q!Γ(2 − β + η) µ,β,η q+2 (q) 1−β |z| |J0,z {z f (z)}| − Γ(2 − β)Γ(2 − µ + η) αq!(p + 2)!Γ(p − β + η + 3) (28) ≤ |z|p−β+2 Γ(p − β + 3)Γ(p − µ + η + 3)[p + 1 − (−1)q α] for z ∈ U if β ≤ 1 and z ∈ U − {0} if β > 1. The inequalities in (28) are obtained by the function f (z) given by (10). Taking β = µ in Theorem 5 and 6, we obtain (1,1)
Corollary 3. If f (z) ∈ Mp µ q+2 (q) |Dz {z f (z)}| − (29)
(1,1)
is in the class Mp q! 1−µ |z| Γ(2 − µ)
(q; α), then
αq!(p + 2)! |z|p−µ+2 Γ(p − µ + 3)[p + 1 − (−1)q α]
≤
for z ∈ U and 0 ≤ µ < 1. The equalities in (29) are attained by the function f (z) given by (7). (1,−1)
Corollary 4. If f (z) ∈ Mp µ q+2 (q) |Dz {z f (z)}| − (30)
≤
(1,−1)
is in the class Mp q! 1−µ |z| Γ(2 − µ)
(q; α), then
αq!(p + 2)! |z|p−µ+2 Γ(p − µ + 3)[p + 1 − (−1)q α]
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for z ∈ U and 0 ≤ µ < 1. The inequalities in (30) are obtained by the function f (z) given by (10). Taking µ = 0 in corollaries 3 and 4 (or β = µ = 0) in Theorems 5 and 6, we obtain (1,1)
Corollary 5. If f (z) ∈ Mp (q; α), then (31) (q) αq! |f (z)| − q!|z|−q−1 ≤ |z|p−q [p + 1 − (−1)q α]
(z ∈ D; p ∈ N, p ≥ q; q ∈ N0 )
The result is sharp for the function f (z) given by (7). (1,−1)
Corollary 6. If f (z) ∈ Mp (q; α), then (q) αq! |f (z)| − q!|z|−q−1 ≤ |z|p−q (32) −q [p + 1 + (−1) α] (z ∈ D; p ∈ N, p ≥ q; q ∈ N0 ). The result is sharp for the function f (z) given by (10). We note that, in their special case q = 0, corollaries 5 and 6 would provide us with the growth properties of functions belonging to the (1,1) (1,−1) classes Mp (q, α), and Mp (q, α), respectively. For q ∈ N0 , these results may be looked as distortion theorems for the classes involved.
4. Meromorphically Starlikeness and Meromorphically (A,B) Convexity of Functions in the Class Mp (q, α) We begin by proving (1,1)
Theorem 7. If f (z) ∈ Mp (q, α), then the function f (z) is meromorphically starlike of order β (0 ≤ β < 1) in |z| < r1 and meromorphically convex of order β (0 ≤ β < 1) in |z| < r2 , where 1 (1 − β)n![n + 1 − (−1)−q α] n+1 (33) r1 = r1 (q; p, α, β) = inf n αq!(n − q)!(n − β + 2) and (34) r2 = r2 (q; p, α, β) = inf n
n > p ≥ q; p ∈ N; q ∈ N0 ).
(1 − β)(n − 1)![n + 1 − (−1)−q α] αq!(n − q)!(n − β + 2)
1 n+1
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Each of these results is sharp for the function f (z) given by (35)
f (z) =
1 αq!(n − q)! + zn. z n!(n + 1 − (−1)−q α
Proof. It is sufficient to show that zf 0 (z) − < 1 − β, (36) − 1 f (z)
(0 < |z| < r1 )
and that (37)
zf 00 (z) − < 1 − β, − 2 f 0 (z)
(0 < |z| < r2 )
(1,1)
for a function f (z) ∈ Mp (q, α), where r1 and r2 are defined by (33) and (34), respectively. The details involved are fairly straightforward and may be omitted. The derivations of Theorem 7 can be applied mutates mutandis in (1,−1) order to obtain the following theorem for the class Mp (q, α). (1,−1)
Theorem 8. If f (z) ∈ Mp (q, α), then the function f (z) is meromorphically starlike of order δ (0 ≤ δ < 1) in |z| < r3 and meromorphically convex of order δ (0 ≤ δ < 1) in |z| < r4 , where 1 (1 − δ)n![n + 1 − (−1)−q α] n+1 (38) r3 = r3 (q; p, α, δ) = inf n αq!(n − q)!(n − δ + 2) and (39) r4 = r4 (q; p, α, δ) = inf n
(1 − δ)(n − 1)![n + 1 − (−1)−q α] αq!(n − q)!(n − δ + 2)
1 n+1
n > p ≥ q; p ∈ N; q ∈ N0 ). Each of these results is sharp for the function f (z) given by (40)
f (z) =
1 αq!(n − q)! − zn. z n![(n + 1 − (−1)−q α]
We note that, meromorphically starlike and meromorphically convexity of functions belonging to several simpler classes can be deducted by suitably choosing the parameter occurring in Theorem 7 to Theorem 8 above.
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Department of Mathematics, Walchand College of Engineering Sangli 416 415, India
