Inequalities for inner radii of symmetric disjoint domains

  • A. K. Bakhtin
  • L.V. Vyhovs'ka
  • I. V. Denega

Abstract

We study the following problem: Let $a_0 = 0, | a_1| = ... = | a_n| = 1,\; a_k \in B_k {\subset C}$, where $B_0, ... ,B_n$ are disjoint domains, and $B_1, ... ,B_n$ are symmetric about the unit circle. It is necessary to find the exact upper bound for $r^{\gamma} (B_0, 0) \prod^n_{k=1} r(B_k, a_k)$, where $r(B_k, a_k)$ is the inner radius of Bk with respect to $a_k$. For $\gamma = 1$ and $n \geq 2$, the problem was solved by L. V. Kovalev. We solve this problem for $\gamma \in (0, \gamma_n], \gamma_n = 0,38 n^2$, and $n \geq 2$ under the additional assumption imposed on the angles between the neighboring line segments $[0, a_k]$.
Published
25.09.2018
How to Cite
BakhtinA. K., Vyhovs’kaL., and DenegaI. V. “Inequalities for Inner Radii of Symmetric disjoint domains”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, no. 9, Sept. 2018, pp. 1282-8, http://umj.imath.kiev.ua/index.php/umj/article/view/1634.
Section
Short communications