Application of upper estimates for products of inner radii to distortion theorems for univalent functions - I. V. Denega; Ya. V. Zabolotnyi (Математичні Студії, 2023)
In 1934 Lavrentiev solved the problem of maximum of product of conformal radii of two non-overlapping simply connected domains. In the case of three or more points, many authors considered estimates of a more general Mobius invariant of the form $$ T_{n}:={\prod\limits_{k=1}^n r(B_{k},a_{k})}{\bigg(\prod\limits_{1\leqslant k<p\leqslant n} |a_{k}-a_{p}|\bigg)^{-\frac{2}{n-1}}}, $$ where $r(B,a)$ denotes the inner radius of the domain $B$ with respect to the point $a$ (for an infinitely distant point under the corresponding factor we understand the unit). In 1951 Goluzin for $n=3$ obtained an accurate evaluation for $T_{3}$. In 1980 Kuzmina showed that the problem of the evaluation of $T_{4}$ is reduced to the smallest capacity problem in the certain continuum family and obtained the exact inequality for $T_{4}$. No other ultimate results in this problem for $n \geqslant 5$ are known at present. In 2021 \cite{Bakhtin2021,BahDen22} effective upper estimates are obtained for $T_{n}$, $n \geqslant 2$. Among the possible applications of the obtained results in other tasks of the function theory are the so-called distortion theorems. In the paper we consider an application of upper estimates for products of inner radii to distortion theorems for univalent functions in disk $U$, which map it onto a star-shaped domains relative to the origin.