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
Bakhtin, A. K., L. Vyhovs’ka, and I. V. Denega. “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