Speaker
Description
Let $\mathbb{C}$ be the complex plane, $\overline{\mathbb{C}}=\mathbb{C}\bigcup\{\infty\}$ be its one point compactification. A function $g_{B}(z,a)$ which is continuous in $\overline{\mathbb{C}}$, harmonic in $B\backslash \{a\}$ apart from $z$, vanishes outside $B$, and in the neighborhood of $a$ has the following asymptotic expansion $$g_{B}(z,a)=-\log|z-a|+\gamma+o(1),\quad z\rightarrow a,$$ is called the (classical) Green function of the domain $B$ with pole at $a\in B$. The inner radius $r(B,a)$ of the domain $B$ with respect to a point $a$ is the quantity $e^{\gamma}$. By using the variational method G.M. Goluzin established that for functions $f_{k}(z)$ which univalently map the disc $|z|<1$ onto mutually non-overlapping domains, $k\in {1,2,3}$, exact estimate holds $$\left|\prod\limits_{k=1}^3 f_{k}'(0)\right|\leq\frac{64}{81\sqrt{3}} |(f_{1}(0)-f_{2}(0))(f_{1}(0)-f_{3}(0))(f_{2}(0)-f_{3}(0))|.$$ Equality is attained only for functions $w=f_{k}(z)$ which conformally and univalently map the disc $|z|<1$ onto the angles $2\pi/3$ with vertex at point $w=0$ and bisectors of which pass through points $f_{k}(0)$, $|f_{k}(0)|=1$. E.V. Kostyuchenko proved that the maximum value of multiplication of inner radiuses for three simply connected non-overlapping domains in the disk is attained for three equal sectors. However, this statement remains valid for multiply connected domains $D_k$. We have considered an extremal problem on the maximum of product of the inner radii on a system of $n$ mutually non-overlapping multiply connected domains $D_k$ containing the points $a_k$, $k = 1, . . . , n$, located on an arbitrary ellipse $\frac{x^{2}}{d^{2}}+\frac{y^{2}}{t^{2}}=1$ for which $d^{2}-t^{2}=1$.
