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

Abstract

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.