Estimates of conformal radius and distortion theorems for univalent functions. (English. Russian original) Zbl 1002.30014

Kovalev, L. V.

J. Math. Sci., New York 110, No. 6, 3111-3120 (2002); translation from Zap. Nauchn. Semin. POMI 263, 141-156 (2000).

Let \(r(D,a)\) be the conformal radius of a simply connected domain \(D\supset\{0,1\}\) with respect to a point \(a\in D\). Considering the function \(r(D,\varepsilon)\) for \(\varepsilon\downarrow 0\) the author gives a simple proof of the recent result of E. G. Emel’yanov [J. Math. Sci., New York 89, No. 1, 976-987 (1998); translation from Zap. Nauchn. Semin. POMI 226, 93-108 (1996; Zbl 0907.30025)] concerning the maximum of \(r(D,1)\) for a family of simply connected domains \(D\) with a fixed value \(r(D,0)=R\). He also solves the similar problem for convex domains. Furthermore, he obtains exact estimates for functionals of the form \(|g'(w)|/|g(w)|^\delta\) which hold for functions \(g\) being inverse to mappings of the known class \(S\) or of the class \(S_M\), respectively, where \(S_M\) denotes the subclass \(\{f\in S: |f(z)|<M\) for \(|z|<1\}\).

Reviewer: Erich Hoy (Freiberg)

MSC:

30C75	Extremal problems for conformal and quasiconformal mappings, other methods
30C55	General theory of univalent and multivalent functions of one complex variable

Keywords:

conformal radius; extremal problem; distortion theorem

Citations:

Zbl 0907.30025