Indian J. pure appl. Math., 27(2) : 165-172,February1996
SOME APPLICATIONS OF DIFFERENTIAL SUBORDINATION 2 AND T. N. SrnNuuceu V. RevIcHANDRANT' Department of Mathematics, Anna University, Madras 600 025
(Received27 April 1995; after revision 22 September1995; accepted6 October 1995)
Some applications of differential subordination are given for certain classes of functions defined through Ruscheweyh derivatives, and for the classesof starlike and convex functions associatedwith the parabolic region Re{w} > | w - 1 l.
L. INrnooucrloN For positive integers p, n, let A,(p) denote the class of all functions of the form
a** fu)=zp+ ; o#*o which are analytic in the unit disk U = {t; | , I . 1}. Denote the class At(p) by A(p) and the class A(1) by A. lret Hfa, nf denote the class of all functions of the form
flz):a+ i a** k=n which are analytic in U where a is a complex number. l-et hQ) be a univatent analytic function with ft(0) = 1. Let / and F be analytic in U. Then / is subordinate to F (written f < F or fQ) < F(r)) if /(0) = r(0) and fltl) EF(U). Denote the 4, h(z) and subclasses of. A,(p) consisting of functions /(z)' for which :+P pNz) " I | " , zf (z)f -., r^t-t k., c? (h\ and f}\ resnectivelv classes.have a respectively'These classes' CVn,p(/r) h(r\ by ST,,o(ft) qnAcv ,b\ ,1, f l< nulurut g"noruirution which are defined through the Ruscheweyh derivatives. Some inclusion results and sufficient conditions are obtained for these classeswhich either extends or improves the earlier results of Chen and Lan1, Miller and Mocano3. @ 2Pr"r"nt
council Scientific and Industrial Research, New Delhi. 600 054' Address : Department of Mathematics, St. Peters Engineering College, Madras
V. RAVICHANDRAN AND T. N. SHANMUGAM
L66 A function fl)
'
- 1 < ct'< 1 if and is uniformly convex of order a with - cr} > lw - 1 l' In other lies in the parabolicregion Re {w eA
only if t *{R t ( ) words, the function flz) is uniformly convex of order cr if
)
t+f\l, * # ' \ z ) < 1 * 2 ( r - s ' ) [ t "Lr -f G f )l' L \ UCV(a)' The class of This class was introduced by RQnningsand it is denoted by lies in the paraboric region is denoted all analytic functions flz) €A for which +P I \z) analytic functions and by So(c). These classes are extended to the class of p-valent some sharp inclusion results for these new classes are obtained' results of Miller and In order to.prove our theorems we need the following Mocanu3. qQ) b" univalent with Theorem A - Lrt a ) 0, n a positive integer, and let q(0) = 1 and qQ) * 0' Set
andh(z) = qQ)+ a n QQ)
QQ)=4P q\z) and suppose that
' o,ze (r andeither (a) \ / ReILPAA atzl I I
|
(b) h@ is convef, or (b') log qQ) is convex (O@ is starlike)' If B € H [], nl and B 4 h, then the analytic solution wP' (z) + B(z) P(z) = | is given bY z
p(z) = a-rrr/cr s-nO I
sPOl/a-t 41
0
where b(z) = s-t
i
tB(r)- rl rt dt,
0
The subordination is sharp. satisfiesp A H ll, nl, P(z)'O and O " I and let q(z) be analytic in U with Theorem B - Let a >0, fl a Positive integer q(0) = L, and set
h(z)=q(z)+*'#))
SOME APPLICATIONSOF DIFFERENTIALSUBORDINATION
167
If (i) Re qQ) > 0, for z e U and (ii) h(z) is convex or log q(z) is convex, then
R"lLIIPI , o rorzeu I erzt J
and both loe4Q)
and qQ) are univalent.
2. Resulrs l-et n Theorem 2.I q(0) = 1 and q(z) * 0. Set
be a positive integer, and let qQ) be univalent with
h ( z' ) = q " '( z ) +pl z qq' \( z?), ) and suppose that qQ) and ft(z) satisfy the conditions (a) and (b) or (a) and (b') of Theorem A. If f F-A^(p), then fe.Cv^,e(ft) implies,f€ ST^,r(e). Pnoop : For the function flz) ecv,,p(ft),
define B(z) by
B (' z) = \ =iLtl .( J t *'4('',)' ! ' . ) '
,l'
G ) 0, Be Hll,nl and B(z) 4 h(z). Note that the functions and let a=l/p.Then qQ) and ft(z) satisfy the conditions of Theorem A with a = L/p. Since z
b(z)= s;t I IBU)- r) rt dt 0
'
r , /
-= u, , Jf ol l l r * t f " ( t \ ) - t l 4 L o l " f , ( , )) - ' J t
- ros - \(t+9\. p
) by an applicationof TheoremA, the analyticsolutionof the differentialequation wp'(z) + B(z\pQ) = | given by z
p(z)=f, {#p-' dt
=ffi
168
V. R.A,VICHANDRAN AND T. N. SHANMUGAM
U q(z). This is equivalentto 1eST,.o@). satisffes* Az) Theorem2.2 - Let cr ) 0, n a positive integer, and let qQ) be univalent with q(0) = L and q(z) - 0. Set
O@-#
andh(z)- q(z)+9 QQ)
and supposethat qQ) and lr(z) satisfy the conditions(a) and O) or (a) and (b') of TheoremA. For geA^(p), definef(z) by
nz)-[ * i s\t)"o".]Then (l feA^(p); (ii) ge,ST,.o(ft)implies.f€ STn.p(q). The resultis sharp.
Pnooror THnoReu 2.2 : Sincegt-,ST,.o(h), the functionB(z)-ffi well=defined, B(z)ehfl,nl
zi
cr{
and B(z) l lr(z). Since
a o l - rnd't-_' "ttol pf u( ) \ " " . t
o)
by TheoremAo the differential equation c rp' (z) + B(z)p(z) = t p has the analytic solution
p{z)--Ir-i o44u" to W* t and LlpQ) < q(z).Sine
'
ilz) = eQ) Mz)lo, a computationshows thar
!.zf'(z) -L zg'(z)'.rszp'(zl
p frz) p sQ) p dz) zP(z) = B(zl -'-' *s P P(z) =
rlris showsthat
;#
I p(r)'
< q@.
In particularwe have the following result due to Miller and Mocanu3.
is
SUBORDINATION SOMEAPPLICATIONS OF DIFFERENTIAL Corollary 2.1 -
\69
l-et q(z\ be univalentwith q(0) = L and qQ) * 0. Set
h(z)=q(z)+r# and suppose that q(z) and ft(z) satisfy the conditions (a) and (b) or (a) and (b') of Theorem A. If f eA,, then /€ CV(h) implies f esf@).
A function flz)eAr@) is p-valent uniformly convex of order B with - 1 . P parabolicregionRe w - p > l, - 11. in the < 1 if and only ff ! ( ^t *' 49) nm ------ r--'(z) "-p f \ ) It is equivalent,; ',
t [ , *+ P ] nI rt r* - . s a [' " * lfi g \ 1 " r-G nz f'(z)
tt^'
I
L'""|,
))'
This class is denotedby UCV,.,(P).The classSP,,'(P) is definedin a similar manner.It is known that UCV CSo. The following result gives the sharp form of the aboveresultfor a more generalclassesof p-valentfunctions.In view of Theorem 2.1 and TheoremB, the following result is obtained. Cprollary 2.2 - Suppose f eUCV,.p(P) and qQ) be definedby ,
=I +z(r;P) q(z) +r rqr!1,) r", - v r ) l ) l" lz) n ' l - \fl -*+ l,
Then {(?\
f(;
4t q(z)'
The result is sharp. For two analyticfunction flz)=il+ 2i-n*pxk* and 8{z)-*+ L|-r*pbp*, their Hadamardproduct or Convolution (f * d Q) is defined by (f * d Q) = zn+Zi-n*o apbp*. For 0 =-p, the Ruscheweyh derivativeof order 0+p - 1 is given by D6*p-t flr)=d*"*
flz).
The following theorem,a natural extensionof the above results to the class of functionsdefinedthroughthe Ruscheweyhderivatives,improvesan earlier result of' Chen and Lanr. Theorem 2.3 - I-et 6,p, a&, realnumbers suchthat0 +p>O E+p+L>o > 0 and n be a positive integer.Supposep(z) be univalentwith p(0) = L and pQ) - 0. Set ,,\ hlzl=
ct a.n zp'(z) 0+p+1-0_r_,, ' 6+p+ 1+-;+ I Az)+ 6+p+ I 6
V. RAVICHANDRANAND T. N. SHANMUGAM
170
and supposethat qQ) and ft(z) satisfy the conditions(a) and (b) or (a) and (b') of TheoremA. If f e A,(p), then p6 *p flz\
+ (1 _ alffii o=r{:::]fu Du*pflz)
4 h(z)
implies tD6 + o {-\
Dr*ft,
Theorem2.4 2.3, and let
Suppose6,p, o, n, e(z),&(z)satisfy the conditionsof Theorem
k(z)=z expi ,nU,- r) rr dt. 0
If /€ A,(A) satisfies T \r -.,r$-ltg a he) "4JlT3+(1 *, D*p D + p-r fl z )
fl z)
where
(z) + n) +n) on h(z\ " o + p + I zk" l+Jt) " - --a(1 0 4 p , + I*" [ 't--s.a(1 +.I k 2 i r t c (r) I then
4.'^r)
< zt((z)
6+e_r fl2,)
Pnoon or THeoneu 2.3 -
Kz)
Detine h1{z)by
t n + 1)lr(z)-cr h{z) \ - ' -(o 0 + p + l - c r Then ft,(z) satisfies the conditions
ht@)_p{z).ffi;# Sincc 6+p+1>c>0, conditions(a) and (b) of TheoremA are satisfiedby hr(zJ whenevcr ft(z) satisfies the conditions. l*;t B(z) be defined by +P t 1) t D'6+P+rflzl ",-64p, + L{::-ffi+(l
n@) --(o
- Offil
*-
SOME APPLICATIONSOF DIFFERENTIALSUBORDINATION
I7I
Then the function B(z) is in H[1, nl and BQ) 4 hrQ). The differential equation
B(z)p(z)+0+t:1 _., zp'(z)= 1 has the solution given by
d4= This can be seendirectly by substitutingp(z) in the differentialequationand using the identity {D6 + o-' flr))' = (6 + p) D6' pf(z)- 0 D6+p r flz). Of course,by using the above identity, we get zp' (z) __z(D6*p flz))'*z(D6*p-r flz))' p(z) Da* P flz) Po * p- r flz)
=- ( o+ p + \ D W + ( o o +l f f i , *
r.
Therefore,
B(z)+ilfl*#-h which shows that p(z) is the solution of the differential equation. Since 6 + p + 1 > cr > 0, Theorem A shows that the function pQ) is in f/[1, n] and d4 * 0 and pQ) < llqQ).This completes the proof. The proof of Theorem 2.4 is similar. Corollary 2.3 (Miller and Mocanu3) L and qQ) * 0. Set 'q' '(?) h' b) \-/ - a(z\ 7\-,f *' q(z)
Let qQ) be univalent in U with q(0) =
'
and supposethat the conditions(a) and (b) or (a) and (b') of TheoremA are satisfied by qQ) and hQ). lt kQ) is as in Theorem2.4 and /€A satisfies 4" Q) n- zk" (z) k'(z) ' f'(z) then
# i s a n a r y t i c a#n.dW
\ \ t72
V. RAVICHANDRAN AND T. N. SHANMUGA,M
RBreneNcss 1. M. chen and L. L-an,Internat. J. Math. & Math. sci. 12 (1) (19g9), 107-lz. 2. S. S. Miller and P. T. Mocanu,MichiganMath. J. 28 (1991), 157-71. 3. S. S. Miller and P. T. Mocanu, Current Topics in Analytic Function Theory (ed. : H. M. Srivastava and S. Owa), World Scientific, 1992, pp. 175-g5. 4. F. Rgnning,Proc. Am. Math. Soc. ll8 (1993),tg9-96. 5. F. Ronning,Ann. univ. Marie curie-sklodowsrta, sect.A, xLV(14) (1991),ll7-zz.
