The sum of coefficients of bounded univalent functions

Abstract

ABSTRACT. We solve the maximal value problem for the functional Re ~jm=z ak i in the class of functions f(z) = z + a2 z ~ +"" 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 large M whenever the set of indices kz, ... , k,~ contains an even number. The Koebe functionsare the boundary functions for many extremal problems in the class S. These functions map the disk E onto the plane cut along the radial ray with slope angle a. The Pick functions pM, defined by the equation