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

##### 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