A converse to the Schwarz lemma for planar harmonic maps

A sharp version of a recent inequality of Kovalev and Yang on the ratio of the $(H^1{)}^{\ast }$ and $H^4$ norms for certain polynomials is obtained. The inequality is applied to establish a sharp and tractable sufficient condition for the Wirtinger derivatives at the origin for harmonic self-maps of the unit disc which fix the origin.