The maximum of the product of conformal radii of nonoverlapping domains in the disk
type: Item
instance of: scholarly article
title: The maximum of the product of conformal radii of nonoverlapping domains in the disk
zbMATH Open document ID: 0940.30011
DOI: 10.1007/BF02172468
author: V. O. Kuznetsov
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-11-22T00:00:00Z
review text: Let \(a\) be a point in the bounded simply connected domain \(D\subset\mathbb{C}\). There exists a unique conformal map \(f\) of \(D\) onto the unit disk in \(\mathbb{C}\) such that \(f(a)= 0\) and \(f'(a)> 0\). The quantity \(R(D,a)= {1\over f'(a)}\) is called the conformal radius of \(D\) with respect to the point \(a\). The author considers some extremal problems. Let us state one of them. Fix \(n\geq 2\) and for \(n\) simply connected not overlapping domains \(a_1\in D_1,\dots, a_n\in D_n\) in the unit disk \(U\subset\mathbb{C}\) consider a functional \[ F(a_1,\dots, a_n; D_1,\dots, D_n)= \prod^n_{j= 1}R(D_j, a_j). \] It is known due to the result of G. Kuz'mina that the only extremal system of domains \(D^*_1,\dots, D^*_n\) (\(a_1,\dots, a_n\) are fixed for a moment) for \(F\) are circular domains of the quadratic differential \[ Q(z)dz^2=- {P(z)\over 4\pi^2R^2(z)} dz^2, \] where \[ R(z)= \prod^n_{j=1} (z- a_j)(1-\overline a_jz),\quad P(z)= B\prod^{2n- 2}_{j= 1} (z- b_j)(1-\overline b_jz), \] where \(B,b_1,\dots, b_{2n-2}\) are uniquely determined by the conditions \(B>0\), \(|b_j|\leq 1\), \(P(a_j)= (R'(a_j))^2\) and \(\bigcup^n_{j=1}\overline D^*_j=\overline U\). Consider now the extremal problem for \(F(a_1,\dots, a_n; D_1,\dots, D_n)\) when \(a_j\) also vary. One of the results of the present paper is the following: Theorem 1. The quadratic differential \(Q(z) dz^2\) defining an extremal system of domains and points has at least two distinct zeroes on the circle \(C=\partial U\). The extremal system here is the system which is the point of maximum of the functional \(F\).
Mathematics Subject Classification ID: 30C75
Mathematics Subject Classification ID: 30C20
Mathematics Subject Classification ID: 30F30
zbMATH DE Number: 1364718
zbMATH Keywords: conformal radius
zbMATH Keywords: quadratic differential
reviewed by: S. Ivashkovich
MaRDI profile type: Publication
cites work: Q4205682
cites work: Geometric properties of extremal decompositions and estimates for the modulus of families of curves in an annulus
full work available at URL: https://doi.org/10.1007/bf02172468
OpenAlex ID: W2094163241
description: scientific article; zbMATH DE number 1364718