A converse to the Schwarz lemma for planar harmonic maps

TL;DR

This paper establishes a sharp inequality relating norms of polynomials and applies it to derive a precise condition for harmonic self-maps of the unit disc fixing the origin, extending the Schwarz lemma in a novel way.

Contribution

It provides a new sharp inequality for polynomial norms and uses it to characterize harmonic self-maps of the unit disc with fixed origin, offering a converse to the Schwarz lemma.

Findings

Derived a sharp inequality between polynomial norms.

Established a precise condition for harmonic self-maps fixing the origin.

Extended the Schwarz lemma to harmonic maps with new bounds.

Abstract

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.