Differential Subordinations for Nonanalytic Functions

Abstract

In the paper by Mocanu (1980), Mocanu has obtained sufficient conditions for a function in the classes $C^{\mathrm{1}}(U)$, respectively, and $C^{\mathrm{2}}(U)$ to be univalent and to map $U$ onto a domain which is starlike (with respect to origin), respectively, and convex. Those conditions are similar to those in the analytic case. In the paper by Mocanu (1981), Mocanu has obtained sufficient conditions of univalency for complex functions in the class $C^{\mathrm{1}}$ which are also similar to those in the analytic case. Having those papers as inspiration, we try to introduce the notion of subordination for nonanalytic functions of classes $C^{\mathrm{1}}$ and $C^{\mathrm{2}}$ following the classical theory of differential subordination for analytic functions introduced by Miller and Mocanu in their papers (1978 and 1981) and developed in their book (2000). Let $\mathrm{\Omega }$ be any set in the complex plane $\mathbb{C}$, let $p$ be a nonanalytic function in the unit disc $U$, $p\in C^{\mathrm{2}}(U),$ and let $\psi (r,s,t;z):{\mathbb{C}}^{\mathrm{3}}\times U\to \mathbb{C}$. In this paper, we consider the problem of determining properties of the function $p$, nonanalytic in the unit disc $U$, such that $p$ satisfies the differential subordination $\psi (p(z),Dp(z),D^{\mathrm{2}}p(z)-Dp(z);z)\subset \mathrm{\Omega }\Rightarrow p(U)\subset \mathrm{\Delta }$.