On Certain Classes of Convex Functions

For real numbers and such that , we denote by the class of normalized analytic functions which satisfy the following two sided-inequality: where denotes the open unit disk. We find some relationships involving functions in the class . And we estimate the bounds of coefficients and solve the Fekete-Szeg？ problem for functions in this class. Furthermore, we investigate the bounds of initial coefficients of inverse functions or biunivalent functions. 1. Introduction Let denote the class of analytic functions in the unit disc which is normalized by Also let denote the subclass of which is composed of functions which are univalent in . And, as usual, we denote by the class of functions in which are convex in . We say that is subordinate to in , written as , if and only if for some Schwarz function such that If is univalent in , then the subordination is equivalent to Definition 1. Let and be real numbers such that . The function belongs to the class if satisfies the following inequality: It is clear that . And we remark that, for given real numbers and , if and only if satisfies each of the following two subordination relationships: Now, we define an analytic function by The above function was introduced by Kuroki and Owa [1], and they proved that maps onto a convex domain conformally. Using this fact and the definition of subordination, we can obtain the following lemma, directly. Lemma 2. Let and . Then if and only if And we note that the function , defined by (7), has the form where For given real numbers and such that , we denote by the class of biunivalent functions consisting of the functions in such that where is the inverse function of . In our present investigation, we first find some relationships for functions in bounded positive class . And we solve several coefficient problems including Fekete-Szeg？ problems for functions in the class. Furthermore, we estimate the bounds of initial coefficients of inverse functions and bi-univalent functions. For the coefficient bounds of functions in special subclasses of , the readers may be referred to the works [2–4]. 2. Relations Involving Bounds on the Real Parts In this section, we will find some relations involving the functions in . And the following lemma will be needed in finding the relations. Lemma 3 (see Miller and Mocanu [5, Theorem ]). Let be a set in the complex plane and let be a complex number such that . Suppose that a function satisfies the condition for all real and all . If the function defined by is analytic in and if then in . Theorem 4. Let , and Then Proof. First of all, we put and