Problems of extremal decomposition of the Riemann sphere

G. V. Kuz'mina

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 17, Tome 276 (2001), pp. 253-275

Résumé

We apply a variant of the method of the extremal metric to some problems concerning extremal decompositions and related problems. Let $\mathbf a=\{a_1,\dots,a_n\}$ be a system of distinct points on $\overline{\mathbb C}$ and let $\mathscr D(\mathbf a)$ be the family of all systems $\mathbb D=\{D_1,\dots,D_n\}$ of nonoverlapping simply connected domains on $\overline{\mathbb C}$ such that $a_k\in D_k, k=1,\dots,n$. Let $$ J(a)=\max\limits_{\mathbb D\subset\mathscr D(\mathbf a)}\biggl\{2\pi\sum_{k=1}^nM(D_k,a_k)-\frac2{n-1}\sum_{1\le k<l\le n}\log|a_k-a_l|\biggr\}, $$ where $M(D_k,a_k)$ is the reduced module of the domain $D_k$ with respect to the point $a_k$. At present, the problem concerning the value $\max\limits_{\mathbf a}J(a)$ was solved completely for $n=2,3,4$. In this work, we continue the previous author's investigations and consider the case $n=5$. In addition, we consider the problem concerning the maximum of the sum $$ \alpha^2\bigl\{M(D_0,0)+M(D_{n+1},\infty)\bigr\}+\sum_{k=1}^nM(D_k,a_k) $$ in the family $\mathscr D(\mathbf a)$ introduced above, where $\mathbf a=\{0,a_1,\dots,a_n,\infty\}$, $a_k$, $k=1,\dots,n$, are arbitrary points of the circle $|z|=1$, and $\alpha$ is a positive number. We prove that if $\alpha/n\le1/\sqrt8$, then the maximum is attained $\alpha$ only for systems of equidistant points of the circle $|z|=1$. For $\alpha/n=1/\sqrt8$, this result was obtained earlier by Dubinin who applied the method of symmetrization. It is shown that if $n\ge2$, where $\alpha/n\ge1/2$ is an even number, then equidistant points of the circle $|z|=1$ do not realize the indicated maximum.