Grunsky inequalities for univalent functions with prescribed Hayman index

Peter Larkin Duren and M. Schiffer

Vol. 131 (1988), No. 1, 105–117

DOI: 10.2140/pjm.1988.131.105

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 f 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 f at prescribed points of the disk.

Mathematical Subject Classification 2000

Primary: 30C50

Secondary: 30C45, 30C70

Milestones

Received: 15 September 1986

Published: 1 January 1988