Application of upper estimates for products of inner radii to distortion theorems for univalent functions

Abstract

In 1934 Lavrentiev solved the problem of maximum ofproduct 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$$T_{n}:={\prod\limits_{k=1}^nr(B_{k},a_{k})}{\bigg(\prod\limits_{1\leqslant k<p\leqslant n}|a_{k}-a_{p}|\bigg)^{-\frac{2}{n-1}}},$$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 showedthat the problem of the evaluation of $T_{4}$ isreduced to the smallest capacity problem in the certain continuumfamily 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 functionsin disk $U$, which map it onto a star-shaped domains relative to the origin.