Inequalities for the inner radii of nonorevlapping domains - Bakhtin, A. K.; Denega, I. V. (Ukrains’kyi Matematychnyi Zhurnal, 2019)
UDC 517.54 We consider the problem of maximum of the functional$$r^γ\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right),$$where $B_{0},…,B_{n}$, $n≥ 2$, are pairwise disjointdomains in $\overline{\mathbb{C}},$ $a_0=0,$ $|a_{k}|=1$,$k=\overline{1,n},$ and $γ\in (0, n]$ ($r(B,a)$ is the innerradius 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 $γ=1$ and by Kuz’mina for$0