Estimates of the inner radii of symmetric non-overlapping domains

Abstract

We consider the problem of estimation of the functional \( In\left(\upgamma \right)={r}^{\upgamma}\left(B\mathrm{o},0\right)\prod \limits_{k=1}^nr\left({B}_k,{a}_k\right), \) where r(B_k; a_k) is the inner radius of a domain B_k relative to the point a_k, under the condition \( {a}_0=0,\mid {a}_k\mid =1,k=\overline{1,n},{a}_k\in {B}_k\subset \overline{\mathbb{C}}, \) where the domains B_k ∩ B_p = ∅ B_k ∩ B_p = ∅, k ≠ p, k, p = \( \overline{0,n}, \) and the domains B_k; k = \( \overline{1,n}, \) possess a symmetry relative to a unit circle. In some partial cases, this problem was solved in [2–5]. The present work is devoted to the study of the problem for \( \upgamma \in \left(1,{n}^{\frac{1}{3}}\right]\;\mathrm{and}\;n\ge 14. \)