On an asymptotically sharp variant of Heinz's inequality.

Abstract

In 1958, E. Heinz obtained a lower bound for ‖∂ x F‖ 2 + ‖∂ y F‖ 2 , where F is a one-to-one harmonic mapping of the unit disc onto itself keeping the origin fixed. Assuming additionally that F is a K-quasiconformal mapping we aim at giving a variant of Heinz's inequality which is asymptotically sharp as K tends to 1. To this end we prove a variant of Schwarz's lemma for such a mapping F.