Inequalities for the inner radii of nonorevlapping domains

  • A. K. Bakhtin
  • I. V. Denega

Abstract

UDC 517.54
We consider the problem of maximum of the functional $$ r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right), $$ where $B_{0},\ldots,B_{n}$, $n\geq 2$, are pairwise disjoint domains in $\overline{\mathbb{C}},$ $a_0=0,$ $|a_{k}|=1$, $k=\overline{1,n},$ and $\gamma\in (0, n]$ ($r(B,a)$ is the inner radius of the domain $B\subset\overline{\mathbb{C}}$ with respect to $a$). Show that it attains its maximum at a configuration of domains $B_{k}$ and points $a_{k}$ possessing rotational $n$-symmetry. This problem was solved by Dubinin for $\gamma=1$ and by Kuz’mina for $0<\gamma< 1$. Later, Kovalev solved this problem for $n\geqslant5$ under an additional assumption that the angles between neighboring linear segments $[0, a_{k}]$ do not exceed $2\pi / \sqrt{\gamma}$. We generalize this problem to the case of arbitrary locations of the systems of points in the complex plane and obtain some estimates for the functional for all $n$ and $\gamma\in (1, n]$.
Published
25.07.2019
How to Cite
Bakhtin, A. K., and I. V. Denega. “Inequalities for the Inner Radii of Nonorevlapping Domains”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, no. 7, July 2019, pp. 996-1002, http://umj.imath.kiev.ua/index.php/umj/article/view/1492.
Section
Short communications