A Certain Subclass of Analytic Functions

In the present paper, we introduce a class of analytic functions in the open unit disc by using the analytic function       2 3 z z     3 3 q z    , which was investigated by Sokół [1]. We find some properties including the growth theorem or the coefficient problem of this class and we find some relation with this new class and the class of convex functions.


Introduction
Let H denote the class of analytic functions in the unit disc


on the complex plane .Let A denote the subclass of H consisting of functions normalized by and The set of all functions f A    that are convex univalent in by K. Recall that a set E is said to be convex if and only if the linear segment joining any two points of E lies entirely in E. Let the function f be analytic univalent in the unit disc on the complex plane with the normalization.Then f maps onto a convex domain E if and only if Robertson introduced in [2], the class  of convex functions of order , which is defined by If , then a function of this set is univalent and if     0 it may fail to be univalent.We denote K K   .Let S be denote the subset of A which is composed of univalent functions.We say that f is subordinate to F in , written as w z , and . The class of convex functions K can be defined in several ways, for example we say that f is convex if it satisfies the condition Many subclass of K have been defined by the condition (1) with a convex univalent function p, given arbitrary, instead of the functions 1 1 z z   .Janowski considered the function p, which maps the unit disc onto a disc in [3,4].An interesting case when the function p is convex but is not univalent was considered in [5].A function p that is not univalent and is not convex and maps unit circle onto a concave set was considered in [1].Now, we shall introduce the class of analytic functions used in the sequel.
Definition 1.1.The function f A  belongs to the class Let the function  be given by (2).We note that where

Re and Im
i i x q e y q e Thus the curve is symmetric with respect to real axis and where .Especially, if 0 , which maps onto the right of line 3 1 . And we note that if

SQ 
Now we shall find some properties of functions in the class   SQ  .
Theorem 2.1.If a function f belongs to the class Proof.Let f be in  .Then there exists an analytic function with From (4) we have Define g and h so that For some g and h such that respectively.And above subordination equations imply that , the modulus of   f z  satisfies the inequality (5).Next, we shall solve some coefficient problem for a special function to be in the class Then n   and we can easily derive that the inequality (6) is equivalent to

The Relations of the Classes SQ and K
It is well-known that the following implication holds: More generally, the above implication ( 7) is can be generalized as following: Evidently, the implication (7) implies the relation . In this chapter, we find some general relation between the classes Let us denote by Q the class of functions f that are analytic and injective on , where and are such that If q is not subordinate to p, then there exist points 0 0 and and there exists a number for which Then by Lemma 3.1, there exist and    since the inequality (3) induces the following inequality: which is a contradiction to the hypothesis.In case 0 1    , using the inequality (8) again, which is a contradiction to the hypothesis, hence , and If we put 1 2   in Theorem 3.2, we can get next Corollary.