A Valency Criterion for Harmonic Mappings

Abstract

Let $$f=h+{\overline{g}}$$ be a sense-preserving harmonic mapping of the closed unit disk $$\overline{{\mathbb {D}}}$$ with a Blashke product dilatation $$B_m= g'/h'$$ of order m. The aim of this paper is to prove that if $$h'$$ has $$p-1$$ zeros, counting multiplicity, in $${\mathbb {D}}$$ and no zeros on $$\partial {\mathbb {D}},$$ and that $$\begin{aligned} {\text {Re}}\left\{ 1+\mathrm{{e}}^\mathrm{it}\frac{h''(\mathrm{{e}}^\mathrm{it})}{h'(\mathrm{{e}}^\mathrm{it})}\right\} >-\frac{1}{2}\sum _{k=1}^m\frac{1-|a_k|}{1+|a_k|}, \end{aligned}$$ where $$a_1,\ldots , a_m$$ are the zeros of $$B_m,$$ then f is $$(m+p-1)$$ -valent. The proof deploys a surface-theoretic technique based on an effective “pasting” procedure. This is an improvement of an earlier result of Bshouty et al. (Proc Am Math Soc 146:1113–1121, 2018) which asserts that if f is a sense-preserving harmonic mapping on $${\mathbb {D}},$$ with dilatation $$z^m$$ that satisfies the inequality $$\begin{aligned} {\text {Re}}\left\{ 1+ z\frac{h''(z)}{h'(z)}\right\} >-\frac{m}{2},\quad z\in {\mathbb {D}}, \end{aligned}$$ then f is $$(m+p)$$ -valent.