Interior radii of symmetric not leaning domains (Q1603055)

Let \(D\) be a domain of the complex sphere \(\overline{{\mathbb C}}\), \(D^*:=\{z:1/\overline{z}\in D\}\) \(g_D(z,z_0)\) denotes Green's function of the domain \(D\) with the pole \(z_0\). The interior radius of \(D\) with respect to \(z_0\) is given by \[ r(D,z_0)= \begin{cases} \exp(\lim_{z\to z_0}(g_D(z,z_0)+\log|z-z_0|)), & z_0\neq\infty;\\ \exp(\lim_{z\to z_0}(g_D(z,z_0)-\log|z|)), & z_0=\infty. \end{cases} \] As the author notes, many extremal problems for classes of analytic functions may be reduced to problems on not leaning domains. The present article is devoted to solving one extremal problem posed in the survey [\textit{V. N. Dubinin}, Russ. Math. Surv. 49, No. 1, 1-79 (1994); translation from Usp. Mat. Nauk 49, No. 1(295), 3-76 (1994; Zbl 0830.30014)]. The following result gives the solution: Theorem. Let \(B_0,\dots,B_n\) (\(n>2\)) be not leaning domains in \(\overline{{\mathbb C}}\); \(a_k\in B_k\), \(k=0,\dots,n\); \(a_0=0\), \(|a_k|=1\), \(k=1,\dots,n\); \(B_k=B_k^*\), \(k=1,\dots,n\). Then \[ \prod_{k=0}^n r(B_k,a_k)\leq {2^{2n+1/n}\over{(n^2-2)^{n/2+1/n}}} ({n-\sqrt{2}\over{n+\sqrt{2}}})^{\sqrt{2}}. \tag{1} \] If, in addition, the domains \(B_k\) possess classical Green's functions, then the equality in (1) is attained if and only if the points \(a_k\) and the domains \(B_k\) are poles and circular domains, respectively, of the quadratic differential \[ Q(z)dz^2=-{(\alpha z)^{2n}+(2n^2-2) (\alpha z)^n+1\over{z^2(( \alpha z)^n-1)^2}} dz^2,\quad |\alpha|=1. \] The article contains only a scheme of the proof. However, the precise references on the necessary results are presented.