Boundary Schwarz lemma for harmonic mappings having zero of order $p$

TL;DR

This paper extends the Schwarz lemma to boundary cases for sense-preserving harmonic mappings with zeros of order p, providing new bounds and generalizations using automorphisms of the unit disk.

Contribution

The paper improves the classical Schwarz lemma for harmonic mappings with zeros of order p and establishes a boundary Schwarz lemma, including a generalization via automorphisms.

Findings

Improved Schwarz lemma for harmonic mappings with zeros of order p

Established boundary Schwarz lemma for such mappings

Generalized results using automorphisms of the unit disk

Abstract

Suppose $w$ is a sense-preserving harmonic mapping of the unit disk $\mathbb{D}$ such that $w(\mathbb{D})\subseteq\mathbb{D}$ and $w$ has a zero of order $p\geq1$ at $z=0$. In this paper, we first improve the Schwarz lemma for $w$, and then, we establish its boundary Schwarz lemma. Moreover, by using the automorphism of $\mathbb{D}$, we further generalize this result.