International Journal of Mathematical Modeling, Simulation and Applications ISSN: 0973-8355 Vol. 2. No. 4, 2009 pp. 490-507

Differential Sandwich Theorems for Linear Operators S. Sivaprasad Kumar, V. Ravichandran and H. C. Taneja Department of Applied Mathematics, Delhi College of Engineering, Delhi-110 042, India 2 Department of Mathematics, University of Delhi, Delhi-110 007, India 3 Department of Applied Mathematics, Delhi College of Engineering, Delhi-110 042, India E-mail: [email protected]; [email protected]; [email protected] 1

ABSTRACT

In this paper, we obtain several differential sandwich type theorems between analytic functions and certain linear operators of some classes of analytic functions which are introduced here by means of these linear operators. Also some special cases are considered. 2000 Mathematics Subject Classification. Primary 30C45, Secondary 30C80. Key words and phrases: Linear operator, differential subordination, superordination, sandwich. 2000 Mathematics Subject Classification. Primary 30C45, Secondary 30C80.

1. INTRODUCTION Let H be the class of functions analytic in D : = {z ΠC : |z| < 1} and H(a, n) be the subclass of H consisting of functions of the form f (z) = a + an zn + an+1z n+1 + ... with H1 : = H(1, 1). Let Ap denote the class of all analytic functions of the form f(z) = zp +



∑ a z (z Œ D) k

k

(1.1)

k= p+1

and let A : = A1. For two functions f(z) given by (1.1) and g(z) = zp +





k=p+1

bk z k, the Hadamard

product (or convolution) of f and g is defined by (f * g) (z) : = zp +



∑abz k k

k

= : (g * f) (z).

(1.2)

k= p+1

For aj Œ C( j = 1, 2, ..., l) and bj Œ C \{0, –1, –2, ... } ( j = 1, 2, ..., m), the generalized hypergeometric function lFm(a1, ... , al; b1, ..., b m ; z) is defined by the infinite series

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lFm (a1, .... al; b1, ....., bm; z) : =

(α 1 )n ... (α l )n zn 1 n ... (β m )n n ! n =0

∑ (β )

(l £ m + 1; l, m Œ N0: = {0, 1, 2, 3, ...}) where (l)n is the Pochhammer symbol defined by

RS T

1, (n = 0); Γ ( λ + n) = λ ( λ + 1) ( λ + 2 ) ... ( λ + n − 1), (n ∈ N : = 1, 2 , 3 ... ). Γ(λ )

(l)n : =

m

r

Corresponding to the function hp(a1, ..., al; b1, ..., bm; z) : = zplFm(a1, ..., al; b1, ..., bm; z), the Dziok-Srivastava operator [7] (see also [20]) Hp(l, m) (a1, ..., al; b1, ..., bm) is defined by the Hadamard product Hp(l, m)(a1, ..., al ; b1,..., bm) f(z) : = hp(a1, ..., al ; b1, ..., bm; z) * f(z) = zp +

(a 1 )n - p ... (a l )n - p an zn (b 1 )n - p ... (b m )n - p (n - p) n= p+1 •

Â

(1.3)

Special cases of the Dziok-Srivastava linear operator includes the Hohlov linear operator [8], The Carlson-Shaffer linear operator Lp (a, c) [4]: Lp(a, c)f(z) = Hp(2, 1), (a, 1; c)f(z).

(1.4)

The Ruscheweyh derivative operator Dn [16]: Dnf(z) =

z( zn − 1 f ( z))(n ) = H1(2, 1) (n + 1, 1; 1) f (z) n!

(1.5)

(n Œ N » {0}; f (z) Œ A), the generalized Bernardi-Libera-Livingston integral operator (cf. [1], [9], [10]) and the SrivastavaOwa fractional derivative operators (cf. [13], [14]). Motivated by the multiplier transformation on A, we define the operator Ip(r, l) on Ap by the following infinite series Ip(r, l)f(z) : = zp +

F k + λI ∑ GH p + λ JK ∞

r

akzk

(l ≥ 0).

(1.6)

k = p+1

( ( The operator Ip(r, l) is closely related to the S a l a gean derivative operators [17]. The operator : = I1(r, l) was studied recently by Cho and Srivastava [6] and Cho and Kim [5]. The operator Ir : = I1(r, 1) was studied by Uralegaddi and Somanatha [21]. For two analytic functions f and F, we say that F is superordinate to f if f is subordinate to F. Recently Miller and Mocanu [12] considered certain second order differential superordinations. ( Using the results of Miller and Mocanu [12], Bulboac a have considered certain classes of first order differential superordinations [3] and superordination-preserving integral operators [2].

ITl

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Over the past few decades, several authors have obtained criteria for univalence and starlikeness depending on bounds of the functionals zf ≤(z)/f(z) and 1 + zf ≤(z)/f¢(z), see Singh [18] and the references therein. In this paper, we not only generalize the results of Ravichandran et al. [15] but also establish the parallel superordination results by obtaining sufficient conditions involving

O( k + 1) f ( z ) O ( k + 2 ) f ( z ) , for functions to satisfy the differential sandwich type O( k ) f ( z) O( k + 1) f ( z ) β

F O( k + 1) f ( z) I p q (z). Also we obtain sufficient conditions involving GH O( k) f ( z) JK O( k + 2) f ( z ) F O( k + 1) f ( z) I F O(k + 1) f (z) IJ p q (z), where O(k) ,G for functions to satisfy q (z) p G J K H z K O( k + 1) f ( z ) H z

implication: q1(z) p

2

δ

δ

1

p

p

2

∫ H pl,m(k) or Ip(k, l) and q1(z), q2(z) are convex univalent functions in D. Also the results involving various special cases of the operators introduced here are discussed.

2. PRELIMINARIES In our present investigation, we need the following: Definition 2.1:

[12, Definition 2, p. 817] Denote by Q, the set of all functions f(z) that are

analytic and injective on D – E(f), where E(f) = {z Œ ∂D : lim f(z) = •}, z→ζ

and are such that f ¢(z) π 0 for z Œ ∂ D – E(f). Lemma 2.1: [11] Let q(z) be univalent in the unit disk D and J and j be analytic in a domain D containing q(D) with j(w) π 0 when w Œ q(D). Set Q(z) : = zq¢(z) j(q(z)), h(z) : = J(q(z)) + Q(z). Suppose that either (i) h(z) is convex, or (ii) Q(z) is starlike univalent in D. In addition, assume that R

zh ′( z) > 0 (z Œ D). Q( z)

If p(z) is analytic in D, with p(0) = q(0), p(D) Œ D and ∂(p(z)) + zp¢(z) j(p(z)) p J(q(z)) + zq¢(z) j(q(z)) = h(z), then p(z) p q(z) and q(z) is the best dominant.

(2.1)

Lemma 2.2: [3] Let q(z) be univalent in the unit disk D and J and j be analytic in a domain D containing q(D). Suppose that 1. R [J¢(q(z))/j(q(z))] > 0 for z Œ D, 2. Q(z) : = zq¢(z) j(q(z)) is starlike univalent in D. If p(z) Œ H[q(0), 1] « Q, with p(D) Œ D, and J(p(z)) + zp¢(z) j(p(z)) is

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univalent in D, then J(q(z)) + zq¢(z) j(q(z)) p J(p(z)) + zp¢(z) j(p(z)),

(2.2)

implies q(z) p p(z) and q(z) is the best subordinant.

3. RESULTS INVOLVING DZIOK-SRIVASTAVA LINEAR OPERATOR We begin with the following: Theorem 3.1: Let a, b and g be real numbers with b π 0 and a1 π –1. Let f(z) Œ Ap, c(z) : =

H lp,m [α 1 ] f ( z ) H lp,m [ α 1 + 1] f ( z )

ΠH1, c(z) and F (z) be defined respectively by

1 {a(a1 + 1) + a1b + [b + g (a1 + 1)] q(z) – bzq ¢(z)} α1 + 1

and

R|α H [α + 1] f ( z) + β H [α + 2] f (z) + γ U| S H [α ] f ( z) H [α + 1] f ( z) V| H [ α + 1] f ( z ) | T W F zq′′(z) I > max RS β + (1 + Rα )γ , 0UV . If F(z) p c(z), then 1. Assume that R G 1 + H q′(z) JK W T β F(z) =

H lp,m [α 1 ] f ( z ) l ,m p

1

l ,m p 1 l ,m p 1

l ,m p l ,m p

1

(3.1)

1

1

H lp,m [ α 1 ] f ( z ) H lp,m [ α 1 + 1] f ( z)

p q(z),

and q(z) is the best dominant. 2. Assume that q(z) be convex univalent in D, q(0) = 1 and b2 + (1 + Ra1)bg < 0. If H lp,m [ α 1 ] f ( z ) H lp,m [ α 1 + 1] f ( z)

q(z) p

ΠQ, F(z) is univalent in D, and c(z) p F(z), then

H lp,m p[α 1 ] f ( z ) H lp,m [ α 1 + 1) f ( z )

and q(z) is the best subordinant. Proof: Define the function y(z) by y(z) =

H lp,m [α 1 ] f ( z )

(3.2)

H lp,m [ α 1 + 1) f ( z )

Then, clearly, y(z) is analytic in D. Also by a simple computation, we find from (3.2) that

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International Journal of Mathematical Modeling, Simulation and Applications z( H lp,m [α 1 ] f ( z )) ′ z( H lp,m [α 1 + 1] f ( z )) ′ zy ¢( z) = – . y ( z) H lp,m [ α 1 ] f ( z ) H lp,m [ α 1 + 1] f ( z )

(3.3)

By making use of the familiar identity z(Hpl, m [a1] f(z))¢ = a1 Hpl, m[a1 + 1] f(z) – (a1 – p)Hpl, m [a1] f(z), equation (3.3) yields ( H pl ,m [α 1 + 2] f ( z ) H lp,m [ α 1

+ 1] f ( z )

=

FG H

α 1 zψ ′ ( z) 1+ 1 − α1 + 1 ψ( z) ψ ( z)

IJ K

(3.4)

(3.5)

By using (3.2) and (3.5), we obtain

LMα H [α + 1] f ( z) + β H [α + 2] f (z) + γ OP H [α ] f ( z) MN H [α ] f ( z) H [α + 1] f (z) PQ H [α + 1] f ( z) L α + β F 1 + α − zψ ′(z) I + γ OP y(z) = M MN ψ(z) α + 1 GH ψ(z) ψ(z) JK PQ l ,m p 1 l ,m p 1

l ,m p l ,m p

l ,m 1 p l ,m 1 p

1

1

1

1

=

1 {a1 + 1)a + a1b + [b + g(a1 + 1)] y(z) – bzy ¢(z)}. α1 + 1

(3.6)

In view of (3.6), the subordination F(z) p c(z) becomes [b + g(a1 + 1)y(z) – bzy¢(z) p [b + g (a1 + 1)q(z) – bzq¢(z) and this can be written as (2.1), where J(w) = [b + g(a1 + 1)]w and j(w) = –b. Note that j(w) π 0 and J(w), j(w) are analytic in C. Let the functions Q(z) and h(z) be defined by Q(z) = zq ¢(z)j(q(z)) = –bzq¢(z), (3.7) h(z) = J(q(z)) + Q(z) = [b + (a1 + 1)g]q(z) – bzq ¢(z). (3.8) In light of the hypothesis stated in Theorem 3.1, we see that Q(z) is starlike and

R

RS zh′( z) UV = R R|S− γ (α T Q( z) W |T

1

F GH

+ 1) + β zq ′′ 1 + 1+ β q ′( z)

I U|V > 0. JK |W

By an application of Lemma 2.1, we obtain that y (z) p q(z) or H lp,m [ α 1 ] f ( z ) H lp,m [ α 1 + 1] f ( z)

p q(z).

The second half of Theorem 3.1 now follows by a similar application of Lemma 2.2. Remark 3.1: Using (1.4), we obtain the result of Ravichandran et al. [15, Theorem 2.1, p. 2221] as a special case to the subordination result of Theorem 3.1.

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Example 3.1 By taking a = 0, the following result is deduced from Theorem 3.1: Let b and g be real numbers with b π 0 and a1 π –1. Let f(z) Œ Ap,

H lp,m [α 1 ] f ( z ) H lp,m [ α 1 + 1] f ( z )

ΠH1, c(z)

and F(z) be defined respectively by c(z) : =

1 {a1b + [b + g(a1 + 1)]q(z) – b zq ¢(z)} α1 + 1

and

R|β H [α + 2] f (z) + γ U| . S H [α + 1] f (z) V| H [ a + 1] f ( z ) | T W F zq ′′( z) I > max RS β + (1 + Rα )γ , 0UV . If F(z) p c(z), then 1. Assume that R G 1 + H q ′( z) JK W T β l ,m p l ,m p

H lp,m [α 1 ] f ( z )

F(z) : =

l ,m p

1

1

(3.9)

1

1

H lp,m [α 1 ] f ( z ) H lp,m [ a 1 + 1] f ( z )

p q(z),

and q(z) is the best dominant. 2. Assume that q(z) be convex univalent in D, q(0) = 1 and b2 + (1 + Ra1)bg < 0. If H lp,m [α 1 ] f ( z ) H lp,m [ a 1 + 1] f ( z )

q(z) p

ΠQ, F(z) is univalent in D, and c(z) p F(z), then

H lp,m [α 1 ] f ( z ) H lp,m [ a 1 + 1] f ( z )

,

and q(z) is the best subordinant. Remark 3.2: Using (1.4), we obtain as a special case the result of Ravichandran et al. [15, Theorem 2.9, p.2227] to the subordination result of Example 3.1. Using Theorem 3.1, we obtain the following differential “sandwich” result: Corollary 3.2: Let a, b and g be real numbers, b π 0 and b2 + (1 + Ra1)bg < 0. Let a1 π –1. Assume that qi(z) (i = 1, 2) be convex univalent in D and qi(0) = 1. Let ci (z) : =

1 {a(a1 + 1) + a1b + [b + g(a1 + 1)] qi(z) – b zq ¢i(z)} α1 + 1

and F(z) be given by (3.9). If f(z) ΠAp,

H lp,m [α 1 ] f ( z ) H lp,m [ α 1 + 1] f ( z )

Œ H1 « Q, F(z) is univalent in D and c1(z)

p F(z) p c2(z), then

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International Journal of Mathematical Modeling, Simulation and Applications H lp,m [ α 1 ] f ( z )

q1(z) p

q2(z),

H lp,m [ α 1 + 1] f ( z)

and q1(z) and q2(z) are respectively the best subordinant and best dominant. By using Theorem 2.1, we obtain the following: Theorem 3.3: Let a, b, g and d be real numbers with a π 0 and a1 π –1. Let f(z) Œ Ap,

FH GH

l ,m p [a 1

+ 1] f ( z)

zp

I JK

d

ΠH1, c(z) and F(z) be defined respectively by 1

c(z) : =

δ( α 1 + 1)

{bd(a1 + 1) + d[(a + g)(a1 + 1)]q(z) + a zq¢(z)}

and

U| F H [α + 1] f (z) I F I z + bG +g JK + 1] f ( z ) z H H [a + 1] f ( z) JK V|W GH F zq ′′( z) I > max RS− δ(α + γ )(Rα + 1 , 0UV . If F(z) p c(z), then 1. Assume that R G 1 + α H q ′( z) JK T W F H [α + 1] f (z) I GH JK p q(z), z R| H F(z) : = Sa |T H

l ,m p [a 1 l ,m p [a 1

+ 2] f ( z )

d

p

l ,m p

l ,m p

δ

1

p

1

(3.10)

1

l ,m p

δ

1

p

and q(z) is the best dominant. 2. Assume that q(z) be convex univalent in D, q(0) = 1 and ab(a + g) (Ra1 + 1) > 0. If

FH GH

l ,m p [α 1

+ 1] f ( z)

zp

FH q(z) p G H

l ,m p [α 1

I JK

δ

ΠQ, F(z) is univalent in D, and c(z) p F(z), then

+ 1] f ( z)

zp

I JK

δ

, and q(z) is the best subordinant.

Proof: Define the function y(z) by

FH y(z) = G H

l ,m p [α 1

+ 1] f ( z)

zp

I JK

δ

(3.11)

where the branch of the function y(z) is so chosen such that y(0) = 1. Then, clearly, y(z) is analytic in D. Also by a simple computation, we find from (3.11) that

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S. Sivaprasad Kumar, V. Ravichandran and H. C. Taneja δz ( H pl ,m [α 1 + 1] f ( z )) ′ zψ ′( z) = – pd. ψ ( z) H lp,m [ α 1 + a ] f ( z )

"'%

(3.12)

By making use of the familiar identity (3.4) in (3.12), we get H lp,m [ α 1 + 2] f ( z ) H lp,m [ α 1

+ 1] f ( z )

=

zY( z) 1 +1 δ(a1 + 1) Y( z)

(3.13)

By using (3.11) and (3.13), we obtain

R| H [α + 2] f ( z) F I S|α H [α + 1] f ( z) + βGH H [α z + 1] f ( z) JK T R| F 1 zy ¢( z) + 1I + b + g U| y(z) = Sa G |T H d(a + 1) y( z) JK y(z) V|W l ,m p l ,+ m p

p

1

l ,m p

1

1

δ

U| F H + γV G |W H

l ,m p [α 1

+ 1] f ( z)

zp

I JK

δ

1

=

1 {bd(a1 + 1) + d(a + g) (a1 + 1)y(z) + azyz)}. δ( α 1 + 1)

(3.14)

In view of (3.14), the subordination F(z) p c(z) becomes d (a + g) (a1 + 1)y(z) + az y ¢(z ) p d(a + g) (a1 + 1)q(z) + azq ¢(z) and this can be written as (2.1), where J(w) = d(a + g) (a1 + 1)w and j(w) = a. Note that j(w) π 0 and J(w), j(w) are analytic in C. Let the functions Q(z) and h(z) be defined by Q(z) = zq¢(z)j(q(z)) = azq¢(z), (3.15) h(z) = J(q(z)) + Q(z) = d(a + g)(a1 + 1)q(z) + azq¢(z). (3.16) By the hypothesis stated in Theorem 3.3, we see that Q(z) is starlike and R

RS zh′( z) UV = R R|S δ(α + γ ) (α α T Q( z) W |T

1

+ 1)

F GH

+ 1+

zq ′′( z) q ′( z)

I U|V > 0. JK |W

Thus, an application of Theorem 2.1, the proof of Theorem 3.3 is completed. Remark 3.3: Using (1.4), we obtain as a special case the result of Ravichandran et al. [15, Theorem 2.5, p. 2224] to the subordination result of Theorem 3.3. Example 3.2:

By taking b = 0 and d = –1, the following result is deduced from Theorem 3.3:

Let a and g be real numbers with a π 0 and a1 π –1. Let f(z) Œ Ap, and F(z) be defined respectively by c(z) : = (a + g) q(z) –

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and

R| S| T

U| z . V + 1] f ( z) |W H [α + 1] f ( z) F zq ′′( z) I > max RS− (α + γ )( Rα + 1 UV . If F (z) p c(z), then 1. Assume that R G 1 + α H q ′( z) JK T W F(z) : = a

H lp,m [α 1 + 2] f ( z) H lp,m [α 1



p

l ,m p

(3.17)

1

1

zp p q(z), + 1] f ( z)

H lp,m [α 1

and q(z) is the best dominant. 2. Assume that q(z) be convex univalent in D, q(0) = 1 and a(a + g) (Ra1 + 1) < 0. If zp Œ Q, F(z) is univalent in D, and c(z) p F(z), then H lp,m [ α 1 + 1] f ( z)

q(z) p

zp , + 1] f ( z)

H lp,m [ α 1

and q(z) is the best subordinant. Remark 3.4: Using (1.4), we obtain as a special case the result of Ravichandran et al. [15, Theorem 2.11, p. 2229] to the subordination result of Example 3.2. Using Theorem 3.3, we obtain the following differential “sandwich” result: Corollary 3.4: Let a, b, g and d be real numbers, a π 0 and ad(a + g)(a1 + 1) > 0. Let a1 π –1. Assume that qi (z) (i = 1, 2) be convex univalent in D and qi(0) = 1. Let ci(z) : =

1 {bd (a1 + 1) + d(a + g) (a1 + 1) qi(z) + azq ¢i(z)} δ( α 1 + 1)

and F(z) be given by (3.17). If f (z) ΠAp,

H lp,m [a1 + 1] f ( z)

c1(z) p F(z) p c2(z), then q1(z) p

zp

(i = 1, 2)

Œ H1 « Q, F(z) is univalent in D and

H lp,m [a1 + 1] f ( z)

p q2(z), zp q1(z) and q2(z) are respectively the best subordinant and best dominant. Theorem 3.5:

F H [ a ] f ( z) I Let a, b, and g be real numbers with b, g π 0. Let f(z) Œ A , G H H [a + 1] f (z) JK p

H1, c(z) and F(z) be defined respectively by

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l ,m p 1 l ,m p 1

b

Œ

S. Sivaprasad Kumar, V. Ravichandran and H. C. Taneja

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c(z) : = aq(z) + gzq¢(z) and

F H [a + 1] f (z) I F(z) : = G H H [a ] f ( z) JK l ,m p 1 l ,m p 1

b

R| L S|β γ MM(α T N

1

+ 1)

H lp,m [α 1 + 2] f ( z) H pl ,m [α 1

+ 1] f ( z)

− α1

H lp,m [α 1 + 1] f ( z) H lp,m [α 1 ] f ( z)

OP PQ

−1 +α

U| V| W

(3.18)

F GH

1. Assume that R 1 +

zq ′′( z ) q ′( z )

F H [α + 1] f (z) I GH H [α ] f ( z) JK l ,m p 1 l ,m p 1

I > max RS− α , 0UV . If F(z) p c(z), then JK T γ W

β

p q(z), and q(z) is the best dominant.

2. Assume that q(z) be convex univalent in D, q(0) = 1 and ag > 0.

F H [α + 1] f (z) I If G H H [α ] f ( z) JK l ,m p 1 l ,m p 1

β

ΠQ, F(z) is univalent in D, and c(z) p F(z),

then

F H [α + 1] f (z) I q(z) p G H H [α ] f ( z) JK l ,m p 1 l ,m p 1

β

and q(z) is the best subordinant.

Proof: Define the function y(z) by

F H [α + 1] f (z) I y(z) = G H H [α ] f ( z) JK l ,m p 1 l ,m p 1

β

.

(3.19)

Then, clearly, y(z) is analytic in D. Also by a simple computation together with the use of the familiar identity (2.1), we find from (3.19) that H lp,m [ a 1 + 2] f ( z ) H lp,m [ α 1 + 1] f ( z) 1 z ψ ′( z ) = (a1 + 1) l ,m – a1 – 1. β ψ ( z) H p [ a 1 + 1] f ( z ) H lp,m [ α 1 ] f ( z )

(3.20)

Therefore, it follows from (3.20) that

F H [α + 1] f (z) I GH H [α ] f ( z) JK l ,m p 1 l ,m p 1

β

R| L S|βγ MM(α T N

1

+ 1)

H lp,m [α 1 + 2] f ( z) H lp,m [a1 + 1] f ( z)

− α1

H lp,m [α 1 + 1] f ( z) H pl ,m [α 1 ] f ( z)

= ay(z) + g zy¢(z).

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OP PQ

−1 +α

U| V| W (3.21)

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International Journal of Mathematical Modeling, Simulation and Applications

In view of (3.21), the subordination F(z) p c(z) becomes ay(z) + azy¢(z) p aq(z) + g zq¢(z) and this can be written as (2.1), where J(w) = aw by

and

j(w) = g.

Note that j(w) π 0 and J(w), j(w) are analytic in C. Let the functions Q(z) and h(z) be defined Q(z) = zq¢(z)j(q(z)) = gzq¢(z), h(z) = J(q(z)) + Q(z) = aq(z) + gzq¢(z). In light of the hypothesis stated in Theorem 3.5, we see that Q(z) is starlike and R

(3.22) (3.23)

RS zh′( z) UV = R R|S α + FG 1 + zq′′( z) IJ U|V > 0. T Q( z) W T| γ H q′( z) K W|

Since J and j satisfy the conditions of Theorem 2.1, the result follows by an application of Theorem 2.1. Remark 3.5: Using (1.4), we obtain as a special case the result of Ravichandran et al. [15, Theorem 2.7, p. 2226] to the subordination result of Theorem 3.5. Using Theorem 3.5, we obtain the following differential “sandwich” result: Corollary 3.6: Let a, b, and g be real numbers and let b, g π 0 and ag > 0. Assume that qi (z) (i = 1, 2) be convex univalent in D and qi(0) = 1. Let ci(z) : = ga qi(z) + zq¢i(z) (i = 1, 2)

F H [α + 1] f (z) I and F(z) be given by (3.18). If f(z) Œ A , G H H [α ] f ( z) JK p

l ,m p 1 l ,m p 1

β

Œ H1 » Q, F(z) is univalent in D and

c1(z) p F(z) p c2(z), then

F H [α + 1] f (z) I q (z) p G H H [α ] f ( z) JK l ,m p 1 l ,m p 1

1

β

p q2(z),

q1(z) and q2(z) are respectively the best subordinant and best dominant.

4. RESULTS INVOLVING MULTIPLIER TRANSFORMATION By making use of Lemmas 2.1 and 2.2, we now prove the following result: Theorem 4.1: Let a, b and g be real numbers with b π 0. Let f(z) Œ Ap,

I p ( r , l ) f ( z) I p ( r + 1 , l ) f ( z)

and F(z) be defined respectively by

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ΠH1, c(z)

S. Sivaprasad Kumar, V. Ravichandran and H. C. Taneja

#

1 {(p + l) (a + b) + g(p + l) q(z) – bzq¢(z)} p+λ

c(z) : = and

|RSα I ( r + 1, λ) f (z) + β I (r + 2, λ) f (z) + γ U|V I ( r + 1, λ ) f ( z) |T I ( r , λ ) f ( z) I ( r + 1 , λ ) f ( z) |W F zq′′(z) I > max RS (p + λ)γ , 0UV . If F(z) p c(z), 1. Assume that R G 1 + H q′(z) JK T β W F(z) : =

I p ( r , λ ) f ( z)

p

p

then

I p ( r , λ ) f ( z) I p ( r + 1 , λ ) f ( z)

p

p

(4.1)

p

p q(z),

and q(z) is the best dominant. 2. Assume that q(z) be convex univalent in D, q(0) = 1 and (p + l)bg < 0. If

I p ( r , λ ) f ( z) I p ( r + 1 , λ ) f ( z)

ΠQ,

F(z) is univalent in D, and c(z) p F(z), then q(z) p

I p ( r , λ ) f ( z) I p ( r + 1, λ ) f ( z)

,

and q(z) is the best subordinant. Proof: Define the function y(z) by y(z) =

I p ( r , λ ) f ( z) I p ( r + 1, λ ) f ( z )

.

(4.2)

Then clearly y(z) is analytic in D. Also by a simple computation, we find from (4.2) that

z( I p ( r , λ ) f ( z) ′ z( I p ( r + 1, λ ) f ( z)′ zp ′( z ) = – . ψ ( z) I p ( r , λ ) f ( z) I p ( r , + 1, λ) f ( z)

(4.3)

By making use of the familiar identity (p + l)Ip(r + 1, l) f(z) = z[Ip(r, l) f(z)]¢ + lIp(r, l)f (z), equation (4.3) yields I p ( r + 2 , λ ) f ( z) I p ( r + 1, λ ) f ( z )

=

FG H

p + l zψ ′( z ) 1 − ψ ( z) p + λ ψ ( z)

(4.4)

IJ K

(4.5)

By using (4.2) and (4.5), we obtain

LMα I (r + 1, λ) f (z) + β I (r + 2, λ) f ( z) + γ OP I ( r , λ) f ( z) MN I (r , λ) f (z) I (r + 1, λ) f (z) PQ I ( r + 1, λ) f ( z) p

p

p

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p

p

p

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#

International Journal of Mathematical Modeling, Simulation and Applications

=

LM a + b F p + l - zy ¢(z) I + g OP y(z) MN y(z) p + l GH y(z) y(z) JK PQ

=

1 {(p + l) (a + b) + g(p + l) y(z) – b zy ¢(z) p+λ

(4.6)

In view of (4.6), the subordination F(z) p c(z) becomes g(p + l)] y(z) – bzp¢(z) p (p + l)q(z) – bzq¢(z) and this can be written as (2.1), where J(w) = g(p + l)w and j(w) = –b. Note that j(w) π 0 and J(w), j(w) are analytic in C. Let the functions Q(z) and h(z) be defined by Q(z) = zq¢(z)j(q(z)) = –bzq¢(z), (4.7) (4.8) h(z) = J(q(z)) + Q(z) = g(p + l)q(z) – bzq¢(z). In light of the hypothesis stated in Theorem 4.1, we see that Q(z) is starlike and R

RS zh′( z) UV = R R|S γ (p + λ) + FG 1 + zq′′( z) IJ U|V > 0. T Q( z) W T| β H q′( z) K W|

By an application of Lemma 2.1, we obtain that y(z) p q(z) or I p ( r , λ ) f ( z) I p ( r + 1 , λ ) f ( z)

p q(z).

The second half of Theorem 3.1 follows by a similar application of Lemma 2.2. Using Theorem 4.1, we obtain the following differential “sandwich” result: Corollary 4.2: Let a, b and g be real numbers, b π 0 and (p + l)bg < 0. Assume that qi(z) (i = 1, 2) be convex univalent in D and qi(0) = 1. Let ci(z) : =

1 {(p + l)(a + b) + g(p + l)qi(z) – bzq¢i(z)} p+λ

and F(z) be given by (4.1). If f(z) ΠAp,

I p ( r , λ ) f ( z) I p ( r + 1 , λ ) f ( z)

(i = 1, 2)

Œ H1 « Q, F(z) is univalent in D and c1(z)

p F(z) p c2(z), then q1(z) p

I p ( r , λ ) f ( z) I p ( r + 1 , λ ) f ( z)

p q2(z),

and q1(z) and q2(z) are respectively the best subordinant and best dominant. Theorem 4.3:

F I ( r + 1 , λ ) f ( z) I Let a, b, g and d be real numbers with a π 0. Let f(z) Œ A , G JK H z p

c(z) and F(z) be defined respectively by

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p

p

δ

ΠH1,

S. Sivaprasad Kumar, V. Ravichandran and H. C. Taneja c(z) : =

#!

1 {b d (p + l) + d[(a + g) (p + l) – p + 1] q (z) + azq ¢(z)} δ( p + λ )

and

R| I ( r + 2, λ) f (z) F z I + γ U| F I ( r + 1, l) f ( z) I + βG F(z) : = Sα JK |T I (r + 1, λ) f (z) H I (r + 1, λ) f (z) JK V|W GH z F zq ′′( z) I > max RS− δ(α + γ ) ( p + λ) , 0UV . If F (z) p c(z), then 1. Assume that R G 1 + α H q ′( z) JK T W F I ( r + 1, λ ) f ( z) I p q(z), GH z JK δ

p

p

p

d

p

p

p

(4.9)

δ

p

p

and q(z) is the best dominant. 2. Assume that q(z) be convex univalent in D, q(0) = 1 and a d(a + g)(p + l) > 0. If δ

F I ( r + 1, λ ) f ( z) I Œ Q, F(z) is univalent in D, and c(z) p F(z), then GH z JK F I ( r + 1 , λ ) f ( z) I q(z) p G JK H z p

p

δ

p

p

and q(z) is the best subordinant. Proof: Define the function y(z) by y(z) =

F I ( r + 1, λ ) f ( z ) I GH z JK p

δ

(4.10)

p

where the branch of the function y(z) is so chosen such that y(0) = 1. Then, clearly, y(z) is analytic in D. Also by a simple computation, we find from (4.10) that

δ z ( I p ( r + 1, λ ) f ( z))′ z ψ ′( z ) = – pd ψ ( z) I p ( r + 1, λ ) f ( z)

(4.11)

By making use of the familiar identity (3.4) in (4.11), we get I p ( r + 2 , λ ) f ( z) I p ( r + 1, λ ) f ( z )

=

z ψ ′( z ) 1 +1 δ( p + λ ) ψ( z)

(4.12)

By using (4.10) and (4.12), we obtain

R| I ( r + 2, λ) f (z) F I S|α I (r + 1, λ) f (z) + βGH I (r + 1z, λ) f (z) JK T p

p

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p

δ



U| F I ( r + 1, λ) f ( z) I V| GH JK z W p

δ

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#"

International Journal of Mathematical Modeling, Simulation and Applications

R| F 1 zψ ′( z) + 1I + β + γ U| y(z) S| GH δ(p + λ) ψ( z) JK ψ(z) V| T W

= α =

1 {bd(p + l) + d(a + g)(p + l)y(z) + a zy(z)} δ( p + λ )

(4.13)

In view of (4.13), the subordination F(z) p c(z) becomes d(a + g)(p + l)y(z) + azy¢(z) p d(a + g)(p + l) q(z) + azq ¢(z) and this can be written as (2.1), where J(w) = d(a + g)(p + l)w and j(w) = a. Note that j(w) π 0 and J(w), j(w) are analytic in C. Let the functions Q(z) and h(z) be defined by Q(z) = zq¢(z)j(q(z)) = azq¢(z), (4.14) (4.15) h(z) = J(q(z)) + Q(z) = d(a + g)(p + l)q(z) + azq¢(z). By the hypothesis stated in Theorem 3.3, we see that Q(z) is starlike and R

R| zh(z) ′ U| = R R| δ(α + l) ( p + λ) F 1 zq′′(z) I U| > 0. +G + S| Q( z) V| S| α H q′(z) JK V|W T W T

Thus, an application of Theorem 2.1, the proof of Theorem 4.3 is completed. Using Theorem 4.3, we obtain the following differential “sandwich” result: Corollary 4.4: Let a, b, g and d be real numbers, a π 0 and ad(a + g)(p + l) > 0. Assume that qi(z) (i = 1, 2) be convex univalent in D and qi(0) = 1. Let ci(z) : =

1 {bd(p + l) + d(a + g)(p + l)qi(z) + azq¢i(z)} δ( p + λ )

and F(z) be given by (4.9). If f (z) ΠAp,

I p ( r + 1, λ ) f ( z)

p F(z) p c2(z), then

zp

(i = 1, 2)

Œ H1 « Q, F(z) is univalent in D and c1(z)

I p ( r + 1 , λ ) f ( z)

q1(z) p

p q2(z), zp q1(z) and q2(z) are respectively the best subordinant and best dominant. Theorem 4.5

F I ( r , λ ) f ( z) I Let a, b, and g be real numbers with b, g π 0. Let f(z) Œ A , G H I ( r + 1, λ) f ( z) JK p

p

p

β

ΠH1,

c(z) and F(z) be defined respectively by c(z) : = a q(z) + gzq¢(z) and F(z) : =

F I ( r + 1 , λ ) f ( z) I GH I ( r , λ ) f ( z) JK p

p

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β

R| L I ( r + 2 , λ ) f ( z ) I ( r + 1 , λ ) f ( z) O U S|βγ ( p + λ ) MM I ( r + 1, λ ) f ( z) − I ( r , λ) f ( z) PP + α |V| N Q W T p

p

p

p

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(4.16)

S. Sivaprasad Kumar, V. Ravichandran and H. C. Taneja

F GH

1. Assume that R 1 +

F I ( r + 1 , λ ) f ( z) I GH I ( r , λ ) f ( z) JK

zq ′′( z ) q ′( z )

##

I > max RS− α , 0UV . If F(z) p c(z), then JK T γ W

β

p

p q(z), and q(z) is the best dominant.

p

2. Assume that q(z) be convex univalent in D and q(0) = 1 and ag > 0. If

F I ( r + 1, λ ) f ( z ) I GH I ( r , λ ) f ( z) JK p

β

Œ

p

Q, F(z) is univalent in D, and c(z) p F(z), then

F Ι ( r + 1, λ ) f ( z ) I q(z) p G H I ( r , λ ) f ( z) JK

β

p

,

p

and q(z) is the best subordinant. Proof: Define the function y(z) by

F I ( r + 1, λ ) f ( z ) I y(z) = G H I ( r , λ ) f ( z) JK

β

p

(4.17)

p

Then, clearly, y(z) is analytic in D. Also by a simple computation together with the use of the familiar identity (2.1), we find from (4.17) that

LM MN

OP PQ

I p ( r + 2 , λ ) f ( z) I p ( r + 1 , λ ) f ( z) 1 zψ ′( z) − = (p + l) . β ψ ( z) I p ( r + 1 , λ ) f ( z) I p ( r , λ ) f ( z)

(4.18)

Therefore, it follows that

F I ( r + 1 , λ ) f ( z) I GH I ( r , λ ) f ( z) JK p

p

β

R| L I (r + 2, λ) f (z) I ( r + 1, λ) f (z) O U S|βγ ( p + λ)MM I ( r + 1, λ) f ( z) − I (r , λ) f (z) PP + α |V| Q W N T p p

p

p

= ay(z) + g zy¢(z) (4.19) In view of (4.19), the subordination F(z) p c(z) becomes ay(z) + g zy¢(z) p aq(z) + gzq¢(z) and this can be written as (2.1), where J(w) = a w and j(w) = g. Note that j(w) π 0 and J(w), j(w) are analytic in C. Let the functions Q(z) and h(z) be defined by Q(z) = zq¢(z) j(q(z)) = gz q¢(z), (4.20) h(z) = J(q(z)) + Q(z) = aq(z) + gzq¢(z). (4.21) In light of the hypothesis stated in Theorem 4.5, we see that Q(z) is starlike and R

Ch10.p65

RS zh ′(z) UV = R R|S a + FG 1 + zq¢¢( z) IJ U|V > 0. T Q(z) W |T g H q¢( z) K |W

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International Journal of Mathematical Modeling, Simulation and Applications

Since J and j satisfy the conditions of Theorem 2.1, the result follows by an application of Theorem 2.1. Using Theorem 4.5, we obtain the following differential “sandwich” result: Corollary 4.6: Let a, b, and g be real numbers and let b, g π 0 and ag > 0. Assume that qi(z) (i = 1, 2) be convex univalent in D and qi(0) = 1. Let ci (z) : = gaqi(z) + zq¢i(z) (i = 1, 2)

F I ( r + 1 , λ ) f ( z) I and F(z) be given by (4.16). If f (z) Œ A , G H I ( r , λ ) f ( z) JK p

p

p

β

Œ H1 « Q, F(z) is univalent in D and

c1(z) p F(z) p c2(z), then

F I ( r + 1 , λ ) f ( z) I q (z) p G H I ( r , λ ) f ( z) JK p

1

p

β

p q2(z),

q1(z) and q2(z) are respectively the best subordinant and best dominant.

REFERENCES 1. S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135 (1969), 429–446. ( 2. T. Bulboac a a, A class of superordination-preserving integral operators, Indag. Math. (N.S.) 13 (2002), no. 3, 301–311. 3. ........., Classes of first-order differential superordinations, Demonstratio Math. 35 (2002), no. 2, 287–292. 4. B. C. Carlson and D. B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal. 15 (1984), no. 4, 737–745. 5. N. E. Cho and T. H. Kim, Multiplier transformations and strongly close-to-convex functions, Bull. Korean Math. Soc. 40 (2003), no. 3, 399–410. 6. N. E. Cho and H. M. and Srivastava, Argument estimates of certain analytic functions defined by a class of multiplier transformations, Math. Comput. Modelling 37 (2003), no. 1-2, 39–49. 7. J. Dziok and H. M. and Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transforms Spec. Funct. 14 (2003), no. 1, 7–18. 8. Ju. E. and Hohlov, Operators and operations on the class of univalent functions, Izv. Vyssh. Uchebn. Zaved. Mat. (1978), no. 10(197), 83–89. 9. R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965), 755–758. 10. A. E. Livingston, On the radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 17 (1966), 352–357.

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#%

11. S. S. Miller and P. T. Mocanu, Differential subordinations, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker Inc., New York, 2000. 12. ..........., Subordinants of differential superordinations, Complex Var. Theory Appl. 48 (2003), no. 10, 815–826. 13. S. Owa, On the distortion theorems. I, Kyungpook Math. J. 18 (1978), no. 1, 53–59. 14. S. Owa and H. M. and Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39 (1987), no. 5, 1057–1077. 15. V. Ravichandran, H. Silverman, S. Sivaprasad Kumar, and K. G. Subramanian, On differential subordinations for a class of analytic functions defined by a linear operator, Int. J. Math. Math. Sci. (2004), no. 41-44, 2219–2230. 16. S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109–115. ( ( 17. G. S. S a al a agean, Subclasses of univalent functions, Complex analysis—fifth RomanianFinnish seminar, Part 1 (Bucharest, 1981), Lecture Notes in Math., vol. 1013, Springer, Berlin, 1983, pp. 362–372. 18. V. and Singh, On some criteria for univalence and starlikeness, Indian J. Pure Appl. Math. 34 (2003), no. 4, 569–577. 19. S. Sivaprasad Kumar, V. Ravichandran, and H. C. Taneja, Differential sandwich results involving certain linear operator, Submitted. 20. H. M. Srivastava, Some families of fractional derivative and other linear operators associated with analytic, univalent, and multivalent functions, Analysis and its applications (Chennai, 2000), Allied Publ., New Delhi, 2001, pp. 209–243. 21. B. A. Uralegaddi and C. Somanatha, Certain classes of univalent functions, Current topics in analytic function theory, World Sci. Publ., River Edge, NJ, 1992, pp. 371– 374.

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Differential Sandwich Theorems for Linear Operators

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