Grunsky inequalities for univalent functions with prescribed Hayman index

Abstract

The Grunsky inequalities in their standard formulation are a generalization of the area principle. Our purpose is to apply a variational method to obtain a stronger system of inequalities which involves both the logarithmic coefficients and the Hayman index of a univalent function / in the usual class S. One immediate consequence is the well-known inequality of Bazilevich on logarithmic coefficients. Another application gives a sharpened form of the Goluzin inequalities on the values of / at prescribed points of the disk.