On the coefficient problem for bounded univalent functions

Abstract

In this paper we shall develop a technique for estimating the coefficients bv (v > 1) of the power series (1) of the class S(bj) which is particularly efficient in the case where b1 is close to' 1. The method is based on a generalization of an inequality, due to Nehari, which represents a necessary and sufficient condition for a function (1) to belong to the class S(bl) [4], [10]. Nehari's inequality is a counterpart of Grunsky's necessary and sufficient condition for an analytic function to be univalent [2]. It is of methodological interest to derive the generalized Nehari inequality in a manner which shows its close relation to the original Grunsky inequality. We will prove it in two different ways. In ?2 we will establish it by the variational method, while in ?3 we shall obtain it by the Grunsky method, using Faber polynomials and their generalizations. In ?4 the results proved will then be applied to the coefficient problem for the class S(b,). We shall treat, in particular, the cases of the coefficients bv with v = 3 and 5. The extension of the considerations for higher indices can be achieved by analogous arguments. To define the terms of our inequality, consider the power series