%0 Journal Article
%T Differential Subordinations of Arithmetic and Geometric Means of Some Functionals Related to a Sector
%A A. Lecko
%A M. Lecko
%J International Journal of Mathematics and Mathematical Sciences
%D 2011
%I Hindawi Publishing Corporation
%R 10.1155/2011/205845
%X Some general theorems on differential subordinations of some functionals connected with arithmetic and geometric means related to a sector are proved. These results unify a number of well known results concerning inclusion relation between the classes of analytic functions built with using arithmetic and geometric means. 1. Introduction For let Let . Let the functions and be analytic in the unit disc . A function is called subordinate to written , if is univalent in and . Let be a domain in and be an analytic function, and let be a function analytic in with and be a function analytic and univalent in . The function is said to satisfy the first-order differential subordination if The general theory of the differential subordinations has been studied intensively by many authors. A survey of this theory can by found in the monograph by Miller and Mocanu [1]. For let It is clear that maps univalently onto the sector of the angle symmetrical with respect to the real axis with the vertex at the origin. In this paper we are interested in the following problem referring to (1.1) to find the constant so that to the following relation is true: with suitable assumptions on function and constants For selected parameters the theorems presented here reduce to the well-known theorems proved by various authors. Particularly, results of this type can be applied to examine inclusion relation between subclasses of analytic functions defined with using arithmetic or geometric means of some functionals, for example, the class of -convex functions or -starlike functions. The lemma below that slightly generalizes a lemma proved by Miller and Mocanu [2] will be required in our investigation. Lemma 1.1 (see [2]). Let be a function analytic and univalent on , injective on and . Let be analytic in . Suppose that there exists a point such that and If and exists, then there exists an for which 2. Main Results In the first theorem which follows directly from Theorem 2.2 [3] we prove that Let us start with the following definition. Definition 2.1 (see [3]). Let and be a function analytic in domain . By will be denoted the class of functions analytic in with and such that the function is well defined in . Theorem 2.2 (see [3]). Let a convex function such that a function analytic in a domain such that , and for . If and then Definition 2.3. Let and By will be denoted the class of functions analytic in of the form (1.4) such that the function is well defined in . Remark 2.4. (1) Setting we see that (2) For each as in Definition 2.3 the class is nonempty. To see this take for
%U http://www.hindawi.com/journals/ijmms/2011/205845/