A unified treatment of some special classes of univalent functions. (English) Zbl 0823.30007

Ma, Wancang; Minda, David

Li, Zhong (ed.) et al., Proceedings of the conference on complex analysis, held June 19-23, 1992 at the Nankai Institute of Mathematics, Tianjin, China. Cambridge, MA: International Press. Conf. Proc. Lect. Notes Anal. 1, 157-169 (1994).

Many authors have been interested in studying various subclasses of univalent functions \(f(z)= z+\cdots\) in the unit disk \(D\). Some of such classes were generated by restricting values of the expressions \({zf'(z)\over f(z)}\) or \({zf'(z)\over f'(z)}+ 1\) to special subdomains of the right half-plane (RHP).
The present authors offer a unified approach to such questions. Let \(\varphi(z)= 1+ a_ 1 z,\dots, a_ 1> 0\) be univalent in \(D\) and map \(D\) onto a domain symmetric about the real axis, situated in the RHP and starlike w.r.t. the point \(\varphi(0)\).
Let \(P(\varphi)\) be the class of all analytic functions \(P(z)= 1+ b_ 1z+\cdots\) in \(D\) and such that \(p(D)\subset \varphi(D)\). The class \(P(\varphi)\) is then used to define clases of convex and starlike families of univalent functions in \(D\). Distortion theorems, covering properties, convolution results and some coefficient inequalities are given.
For the entire collection see [Zbl 0816.00023].

Reviewer: E.Złotkiewicz (Lublin)

MSC:

30C45	Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C75	Extremal problems for conformal and quasiconformal mappings, other methods

Keywords:

distortion theorems; covering properties; convolution results; coefficient inequalities