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JOURNALS // Zapiski Nauchnykh Seminarov POMI // Archive

Zap. Nauchn. Sem. POMI, 2000 Volume 263, Pages 49–69 (Mi znsl1135)

This article is cited in 5 papers

Distortion of the hyperbolic Robin capacity under conformal mapping and extremal configurations

B. Dittmara, A. Yu. Solyninb

a Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg
b St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: This paper is connected with recent results of Duren and Pfaltzgraff (J. Anal. Math., 78, 205–218 (1999)). We consider the problem on the distortion of the hyperbolic Robin capacity $\delta_h(A,\Omega)$ of the boundary set $A\subset\partial\Omega$ under a conformal mapping of a domain $\Omega\subset U$ into the unit disk $U$. It is shown that, for sets consisting of a finite number of boundary arcs or complete boundary components, the inequality
\begin{equation} \operatorname{cap}_hf(A)\ge\delta_h(A,\Omega) \tag{1} \end{equation}
is sharp in the class of conformal mappings $f\colon\Omega\to U$ such that $f(\partial U)=\partial U$. Here $\operatorname{cap}_hf(A)$ is the hyperbolic capacity of a compact set $f(A)\subset U$. We give some examples demonstrating properties of functions which realize the case of equality in relation (1).

UDC: 517

Received: 15.02.1999
Revised: 11.10.1999


 English version:
Journal of Mathematical Sciences (New York), 2002, 110:6, 3058–3069

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