The boundary distortion and extremal problems in certain classes of univalent functions

A. Yu. Solynin

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 11, Tome 204 (1993), pp. 115-142

Résumé

We consider the class $S_1(\tau)$, $0<\tau<1$, of functions $f(z)=\tau z+a_2z^2+\dots$ that are regular and univalent in the unit disk $U$ and have $|f(z)|<1$. We obtain sharp estimates for the $1$-measure of the sets $\{\theta\colon|f(e^{i\theta})|=1\}$. As a corollary, for the familiar class $S$ we find Kolmogorov-type estimates for the sets $\{\theta\colon|f(e^{i\theta})|>M\}$, $M>1$, and prove inequalities for the harmonic measure, which are similar to those by Carleman–Milloux and Baernstein. We also consider problems on distortion of fixed systems of boundary arcs in the classes of functions that are regular (or meromorphic) and univalent in the disk or circular annulus. Bibliography: 22 titles.