On Conformal Radii of Non-Overlapping Simply Connected Domains

The paper deals with the following open problem stated by V.N. Dubinin. Let $a_{0}=0$, $|a_{1}|=\ldots=|a_{n}|=1$, $a_{k}\in B_{k}\subset \overline{\mathbb{C}}$, where $B_{0},\ldots, B_{n}$ are disjoint domains. For all values of the parameter $\gamma\in (0, n]$ find the exact upper bound for $r^\gamma(B_0,0)\prod\limits_{k=1}^n r(B_k,a_k)$, where $r(B_k,a_k)$ is the conformal radius of $B_k$ with respect to $a_k$. For $\gamma=1$ and $n\geqslant2$ the problem was solved by V.N. Dubinin. In the paper the problem is solved for $\gamma\in (0, \sqrt{n}\,]$ and $n\geqslant2$ for simply connected domains.The paper deals with the following open problem stated by V.N. Dubinin. Let a0 = 0, ιa1ι =...= ιanι = 1, ak ∈ Bk ⊂ , where B0, ..., Bn are disjoint domains. For all values of the parameter γ∈ (0; n] find the exact upper bound nfor rγ(B0; 0) ∏ r(Bk; ak), where r(Bk; ak) is the conformal radius of Bk with respect to ak. For γ = 1 k=1 and n ≥ 2 the problem was solved by V.N. Dubinin. In the paper the problem is solved for γ ∈ (0; √n ] and n ≥ 2 for simply connected domains.