A Valency Criterion for Harmonic Mappings

Abstract

Let f= h+ g¯ be a sense-preserving harmonic mapping of the closed unit disk D¯ with a Blashke product dilatation Bm= g′/ h′ of order m. The aim of this paper is to prove that if h′ has p- 1 zeros, counting multiplicity, in D and no zeros on ∂D, and that Re{1+eith′′(eit)h′(eit)}>-12∑k=1m1-|ak|1+|ak|,where a1, … , am are the zeros of Bm, 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 D, with dilatation zm that satisfies the inequality Re{1+zh′′(z)h′(z)}>-m2,z∈D,then f is (m+ p) -valent. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.