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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.doi:10.1016/j.jmaa.2020.124908 fatcat:y3353hvwtne5tjoa33lwbj5j5m