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

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.