Differential subordinations for nonanalytic functions. (English) Zbl 1472.30012

Oros, Georgia Irina; Oros, Gheorghe

Abstr. Appl. Anal. 2014, Article ID 251265, 9 p. (2014).

Summary: In the paper [Math. Rev. Anal. Numér. Théor. Approximation, Math. 22(45), 77–83 (1980; Zbl 0457.30038)], 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)], 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 S. S. Miller and 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 \(\psi(p(z), D p(z), D^2 p(z) - D p(z); z) \subset \Omega \Rightarrow p(U) \subset \Delta\).

MSC:

30C80

Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination

Keywords:

non-analytic functions; subordination

Citations:

Zbl 0457.30038; Zbl 0481.30014; Zbl 0367.34005; Zbl 0439.30015; Zbl 0954.34003

References:

[1]	Mocanu, P. T., Starlikeness and convexity for non-analytic functions in the unit disc, Mathematica, 22, 1, 77-83 (1980) · Zbl 0457.30038
[2]	Mocanu, P. T., Sufficient conditions of univalency for complex functions in the class C1, Revue d’Analyse Numerique et de Theorie de l’Approximation, 10, 1, 75-79 (1981) · Zbl 0481.30014
[3]	Miller, S. S.; Mocanu, P. T., Second order differential inequalities in the complex plane, Journal of Mathematical Analysis and Applications, 65, 298-305 (1978) · Zbl 0367.34005
[4]	Miller, S. S.; Mocanu, P. T., Differential subordinations and univalent functions, Michigan Mathematical Journal, 28, 157-171 (1981) · Zbl 0439.30015
[5]	Miller, S. S.; Mocanu, P. T., Differential Subordination. Theory and Applications (2000), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 0954.34003