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

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 $p\ge 1$ 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.

10 pages; To appear in Bulletin of the Malaysian Mathematical Sciences Society

References in corpus (1)

• The Schwarz type Lemmas and the Landau type theorem of mappings satisfying Poisson's equations