ON SUBORDINATION FOR CERTAIN SUBCLASS OF ANALYTIC FUNCTIONS

In the present paper the class P,[a, M] consisting offunctions f(z)= z+ , akzk(n > 1), k=n+l which are analytic in the unit disc E (z Izl < 1) and satisfy the condition If(z)+ azf"(z) 11 < M is introduced. By using the method of differential subordination the properties of the class P, [a, M] are discussed.


INTRODUCTION
Let A(n >_ 1) denote the class of functions of the form f(z) z + , ak zk which are analytic in k=n+l the unit disc E {z-Izl < 1}.A function f(z) in A, is said to be in P,[c,M] for some a(c > 0) and M(M > 0) if it satisfies the condition If(z) + zf"(z) 1 < M (z ).
( ) Let f(z) and g(z) be analytic in E. Then we say that the function g(z) is subordinate to f(z) in E if there exists an analytic function w(z) in E such that Iw(z)l < 1 (z E) and g(z) f(w(z)) For this relation the symbol g(z) -< f(z) is used.In case f(z) is univalent in E we have that the subordination g(z) -< f(z) is equivalent to g(0) f(0) and g(E) C f(E).
In this paper, we shall use the method of differential subordination [2] to obtain certain properties of the class P,, [a, M].

MAIN RESULTS
In order to give our main results, we need the following lemma.
Further, using (2.1) and (2.2) we can arrive at (2.3) and (2 4) by integration, as follows we can show that all estimates ofthis theorem are sharp.
According to the proof of Theorem 1, we have COROLLARY.Let f(z (1 + ,)( + ,) The results are sharp.
THEOREM 2. Let f(z) P[c,M].IfM < 1 +ha, thenRe{e'af'(z)} > O(z E), where is real and I1 5 arc sin 1+ Izl".The result is sharp in the sense that the range of cannot be increased.

Izl
The result is sharp and the extremal function has the form of (2.8) THEOREM 3. Let f(z) P,[t,M] IfM <_ v/+(+) then f(z)is univalent starlike in E PROOF.According to the corollary and the assumption of Theorem 3, it follows immediately that ey'() > 0( e )d e -> 0( ) On the other hand, we see that M l+n larg f'(z)l < arc sin < arc sin and arg f(z) < arc sin < arc sin (2.12) Using (2.11) and (2.12), we obtain ()l + rg l+n arc sin + arc sin v/l+ ( +,)   (z 6_ E), J1 + (1 -+-n) which implies that f(z) is univalent starlike in E.  which shows that F(z) 5 P,, c+-", This result is sharp and the extremal function has the form of (2.8).
TttEOREM 5. Let c > 1 and a > 0. If F(z) e Pn[a,M], then the function f(z) defined by (2.13) satisfies If'(z) 11 < M for z e E.
The result is sharp.