A converse to the Schwarz lemma for planar harmonic maps

Abstract A sharp version of a recent inequality of Kovalev and Yang on the ratio of the ( H 1 ) ⁎ 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.

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