Differential Subordinations for Non-analytic Functions

Georgia Irina Oros; Gheorghe Oros

Georgia Irina Oros Department of Mathematics and Computer Science, University of Oradea, Universitatii Street, No. 1, 410087 Oradea, Romania.
Gheorghe Oros Department of Mathematics and Computer Science, University of Oradea, Universitatii Street, No. 1, 410087 Oradea, Romania.

Keywords: Differential subordination, non-analytic function, dominant, best dominant

Abstract

In paper [1], Petru T. Mocanu has obtained su?cient conditions for a function in the classes C (U), respectively C 2 (U) to be univalent and to map U onto a domain which is starlike (with respect to origin), respectively convex. Those conditions are similar to those in the analytic case. In paper [2], Petru T. Mocanu has obtained su?cient 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 have tried to introduce the notion of subordination for non-analytic functions of classes C and C 2 following the classical theory of di?erential subordination for analytic functions introduced by S.S. Miller and P.T. Mocanu in papers [3] and [4] and developed in the book [5]. Let ? be any set in the complex plane C, let p be a non-analytic function in the unit disc U, p ? C (U) and let ?(r, s, t; z) : C 3 ? C. In article [6] we have considered the problem of determining properties of the function p, non-analytic in the unit disc U, such that p satis?es the di?erential subordination. ?(p(z), Dp(z), D 2 p(z) ? Dp(z); z) ? ? ? p(U) ? ?. The present chapter is based on the results contained in paper [7], some parts of it have been removed and results obtained after the appearance of the paper have been added.