The maximum of the product of conformal radii of nonoverlapping domains in the disk
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 \(F\).
scientific article; zbMATH DE number 1364718