On the Heins theorem

DOI10.4064/SM181214-8-4zbMATH Open1461.30065arXiv1812.01728OpenAlexW2995077739MaRDI QIDQ5220152

Aleksei Kulikov

Publication date: 10 March 2020

Published in: Studia Mathematica (Search for Journal in Brave)

Abstract: It is known that the famous Heins Theorem (also known as the de Branges Lemma) about the minimum of two entire functions of minimal type does not extend to functions of finite exponential type. We study in detail pairs of entire functions $f,g$ of finite exponential type satisfying $su{p}_{zinmathbbC}min|f(z)|,|g(z)|<infty.$ It turns out that $f$ and $g$ have to be bounded on some rotating half-planes. We also obtain very close upper and lower bounds for possible rotation functions of these half-planes.

Full work available at URL: https://arxiv.org/abs/1812.01728

zbMATH Keywords

harmonic measureextremal lengthgrowth of entire functions

Mathematics Subject Classification ID

Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane (30E20) Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions (31A15) Special classes of entire functions of one complex variable and growth estimates (30D15)

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