Angular derivatives of bounded univalent functions and extremal partitions of the unit disk

Abstract

We study bounded univalent functions f(z) that map the unit disk into itself such that f(0) = 0 and the angular limits f(ζ κ ) with the angular derivatives f'(ζ κ ) exist at fixed points ζ k of the unit circle, k = 1,..., n. We use a general inequality of Schiffer-Tammi type obtained earlier by the authors and discuss the cases of the equality sign. Sharp estimates of functionals are obtained in classes of such functions. An explicit form of extremal functions is deduced. Since one of the methods of solution is based on the extremal partition of the unit disk, we are also concerned with some geometric problems. In particular, we study the problem of the maximum of the sum of the reduced moduli of digons and circular domains. As a corollary we derive sharp estimates of functionals dependent on (|f'(0)|, |f'(ζ 1 )f'(ζ 2 )|).