The sum of coefficients of bounded univalent functions

We solve the maximal value problem for the functional\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\operatorname{Re} \sum\nolimits_{j = 1}^m {a_{k_j } } $$ \end{document} in the class of functionsf(z)=z+a2z2+… that are holomorphic and univalent in the unit disk and satisfy the inequality |f(z)|<M. We prove that the Pick functions are extremal for this problem for sufficiently largeM whenever the set of indicesk1,…,km contains an even number.