Special Issue in honor of Alan C. Lazer. Electron. J. Diff. Eqns., Special Issue 01 (2021), pp. 269-278.

The mean value property and zeros of holomorphic functions (Gauss, Poisson, Bolzano, and Cauchy meet in the complex plane)

Jean Mawhin

Abstract:
An existence condition for a zero of holomorphic functions in a disk is stated and proved in a very simple way using the mean value property. It contains as special cases Bolzano's theorem and Brouwer fixed point theorem in a disk for holomorphic functions, the fundamental theorem of algebra and an asymptotic condition for the existence of zeros of transcendental entire functions. An elementary proof of the used mean value property is given.

Published December 27, 2021.
Math Subject Classifications: 12D05, 30C15.
Key Words: Mean value property; holomorphic functions; Bolzano's theorem; fundamental theorem of algebra.
DOI: https://doi.org/10.58997/ejde.sp.01.m2

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Jean Mawhin
Institut de Recherche en Mathématique et Physique
Université Catholique de Louvain
Chemin du Cyclotron, 2, 1348
Louvain-la-Neuve, Belgium
email: jean.mawhin@uclouvain.be

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