Grunsky inequalities and quasiconformal extension

Abstract

The Grunsky coefficient inequalities play a crucial role in various problems and are intrinsically connected with the integrable holomorphic quadratic differentials having only zeros of even order. For the functions with quasi-conformal extensions, the Grunsky constant ℵ(f) and the extremal dilatationk(f) are related by ℵ(f)≤k(f). In 1985, Jurgen Moser conjectured that any univalent functionf(z)=z+b 0+b 1 z −1+… on Δ*={|z|>1} can be approximated locally uniformly by functions with ℵ(f)<k(f). In this paper, we prove a theorem confirming Moser’s conjecture, which sheds new light on the features of Grunsky coefficients.