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

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. For more information see  https://ejde.math.txstate.edu/special/01/m2/abstr.html