Coefficients of univalent harmonic mappings

2017/03/07 by Ponnusamy, Saminathan, Kaliraj, Anbareeswaran Sairam, Starkov, Victor V. · 1 citation
#30C50 #30C55 #Complex Variables (math.CV) #FOS: Mathematics #Primary: 31A05 #Secondary: 30C45

paper · doi:10.48550/arxiv.1703.02371

Abstract

Let S_H⁰ denote the class of all functions f(z)=h(z)+g(z)=z+∑^∞_n=2 a_nzⁿ +∑^∞_n=2 b_nzⁿ that are sense-preserving, harmonic and univalent in the open unit disk |z|<1. The coefficient conjecture for S_H⁰ is still open even for |a₂|. The aim of this paper is to show that if f=h+g ∈ S⁰_H then |a_n| < 5.24 × 10^-6 n¹⁷ and |b_n| < 2.32 × 10^-7n¹⁷ for all n ≥ 3. Making use of these coefficient estimates, we also obtain radius of univalence of sections of univalent harmonic mappings.

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