Differential subordinations for nonanalytic functions (Q1722442)

Summary: In the paper [Math. Rev. Anal. Numér. Théor. Approximation, Math. 22(45), 77--83 (1980; Zbl 0457.30038)], \textit{P. T. Mocanu} has obtained sufficient conditions for a function in the classes \(C^1(U)\), respectively, and \(C^2(U)\) 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 [Math. Rev. Anal. Numér. Théor. Approximation, Anal. Numér. Théor. Approximation 10, 75--79 (1981; Zbl 0481.30014)], \textit{P. T. Mocanu} has obtained sufficient conditions of univalency for complex functions in the class \(C^1\) 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 \(C^1\) and \(C^2\) following the classical theory of differential subordination for analytic functions introduced by \textit{S. S. Miller} and \textit{P. T. Mocanu} in their papers [J. Math. Anal. Appl. 65, 289--305 (1978; Zbl 0367.34005); Mich. Math. J. 28, 157--171 (1981; Zbl 0439.30015)] and developed in their book [Differential subordinations: theory and applications. New York, NY: Marcel Dekker (2000; Zbl 0954.34003)]. Let \(\Omega\) be any set in the complex plane \(\mathbb{C}\), let \(p\) be a nonanalytic function in the unit disc \(U\), \(p \in C^2(U)\), and let \(\psi(r, s, t; z) : \mathbb{C}^3 \times U \rightarrow \mathbb{C}\). 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 \(\psi(p(z), D p(z), D^2 p(z) - D p(z); z) \subset \Omega \Rightarrow p(U) \subset \Delta\).