Study of simply connected domain and its geometric properties

ABSTRACT By using composition of a differential operator and a subclass of analytic functions, we introduce a new application of a differential operator for starlike and convex functions. Moreover, we are dealing with starlikeness and convexity properties of hypergeometric and related functions.


Introduction
The Geometric Function Theory (GFT) deals with the geometric properties of analytic functions of a complex variable. Geometrically, analytic functions are divided into starlike, convex and close to convex functions. The open unit disk plays an important role in the construction of simply connected domains including starlike and convex domains. The fundamental rule of open unit disk is applied in the foundation of Riemann mapping theorem. It is a classical result in GFT , which states that for every non-empty simply connected (not contain any hole) open subset of the complex plane which is not all of the complex plane, then there exists one-to-one, onto and holomorphic mapping (implies conformal map and angle-preserving) from onto U (the open unit disk). The differential operators defined in the open unit disk has attracted the attention of many researchers.
We are motivated by the research works based on mapping properties of hypergeometric functions, convolutions of starlike and convex functions (cf. [1]). This article provides an idea to introduce a new subclass of analytic functions with the help of differential operator given by (2).
The p-valent functions analytic and univalent in U = {z ∈ C : |z| < 1} of the form (1) with p ∈ N are said to form the class A(p).
A function f of the form (1) is called starlike function of We denote the class of starlike functions of order ξ by S * (ξ , p).
We denote the class of convex functions of order ξ by C(ξ , p). The class S * (ξ , p) was introduced by Patil and Thakare [2] and C(ξ , p) was introduced by Owa [3].