Grunsky inequalities for univalent functions with prescribed Hayman index (Q1078714)

Let S be the usual class of functions f which are analytic and univalent in the unit disk, with \(f(0)=0\) and \(f'(0)=1\). The Grunsky inequalities in their standard formulation are a generalization of the area principle. The authors apply a variational method to obtain a stronger system of inequalities involving both the logarithmic coefficients and the Hayman index of a function f in the class S. One immediate consequence is the well-known inequality of Bazilevich which estimates the logarithmic coefficients in terms of the Hayman index. Another application gives a sharpened form of the Goluzin inequalities on the values of f at prescribed points of the disk.