ON RADII OF CONVEXITY AND STARLIKENESS OF SOME CLASSES OF ANALYTIC FUNCTIONS

Let P[A,B], −1≤B<A≤1, be the class of functions p such that p(z) is subordinate to 1

(1.1) A function g, analytic in E, is called subordinate to a function G if there exi.sts a Schwarz function w(z), w(z) analytic in E with w(0) 0 and w(z)l < 1 in E, such that g(z) G(w(z)).
In [1], Janowski introduced the class P[A,b].For A and B,-1 <B <A 1, a function p, analytic in I+A E with p(0) 1 belongs to the class P[A,B ifp(z) is subordinate to --;-,.
This result is sharp for the function f0 e S*[A,B such that This gives us (z/'(z))' /'(z) Applying the usual inequalities, we obtain Hence we obtain the required result that f e C(_-) for Izl < ro and ro is given by (3.1).
Similarly we can define the class V,[A,B as follows.A function f, analytic in E and given by (1.1) belongs to V[A ,B ], k :,. 2, if and only if -B Re P(Z)lP(z)l +Br The following is the extension of Libera's result [6].LEMMA 2.2.Let N and D be analytic in E, D map onto a many-sheeted starlike region.
PROOF.LetJ(z) /t"-t(F(t))"dt and so and or ((z))".This proves our result.Similarly, we can prove the following: THEOREM 3.7.Let a and m be positive integers and f V[A,B ].Let Fbe defined by (3.3).Then f e C(--'s) for Izl < rl where r is given by (3.2).
Similarly, we have the following: TItEOREM 3.9.Let f and g e V[A,B and, for it, m positive integers, let F be defined by (3.4).

I-A
Then F e C(_---) for [z[ <ro, where ro is as given in Theorem 3.8.
" P[A,B] and let F be defined by THEOREM 3.10.Let g e V[A,B and e Z. Let g e. S*[A,B and let e P[A,B ].Then f e C(_---) for zl < ro, where ro is given by (3.1).PROOF.zf'(z) g(z)p(z), pe P[A,B ].
2) and(1.3), it is clear that f e Vk[A,B if and only if zf' e Rk[A,B (1.4)It may be noted that V2[A,B C[A,B and V,[1,-1] V,, the class of functions of bounded rotation first discussed byPaatero [4].2.PRELIMINARY RESULTS LEMMA 2.1[5] Let p e P[A,B] 0 D(0) and e P[A,B ].Then D-z) e P[A,B ].For the proof of this result we refer to[5].LEMMA 2.3.[7]Let p e P[A,B ].Then, for z e E, a 0 and a 0, we have where and This result is sharp.3.MAIN RESULTS.zp'(z)Re{ aP(Z) + fi p(z) } ct-{A-B) + 2aA}r + ctAUr (I -Ar)(1 -Br)