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April 2020 Conservation of the number of zeros of entire functions inside and outside a circle under perturbations
Michael Gil’
Rocky Mountain J. Math. 50(2): 583-588 (April 2020). DOI: 10.1216/rmj.2020.50.583

Abstract

Let $f$ and $\stackrel{̃}{f}$ be entire functions of order less than two, and $\mathrm{\Omega }=\left\{z\in ℂ:|z|=1\right\}$. Let ${i}_{in}\left(f\right)$ and ${i}_{out}\left(f\right)$ denote the numbers of the zeros of $f$ taken with their multiplicities located inside and outside $\mathrm{\Omega }$, respectively. Besides, ${i}_{out}\left(f\right)$ can be infinite. We consider the following problem: how “close” should $f$ and $\stackrel{̃}{f}$ be in order to provide the equalities ${i}_{in}\left(\stackrel{̃}{f}\right)={i}_{in}\left(f\right)$ and ${i}_{out}\left(\stackrel{̃}{f}\right)={i}_{out}\left(f\right)$? If for $f$ we have the lower bound on the boundary, that problem sometimes can be solved by the Rouché theorem, but the calculation of such a bound is often a hard task. We do not require the lower bounds. We restrict ourselves by functions of order no more than two. Our results are new even for polynomials.

Citation

Michael Gil’. "Conservation of the number of zeros of entire functions inside and outside a circle under perturbations." Rocky Mountain J. Math. 50 (2) 583 - 588, April 2020. https://doi.org/10.1216/rmj.2020.50.583

Information

Received: 30 May 2019; Revised: 3 August 2019; Accepted: 23 October 2019; Published: April 2020
First available in Project Euclid: 29 May 2020

zbMATH: 07210980
MathSciNet: MR4104395
Digital Object Identifier: 10.1216/rmj.2020.50.583

Subjects:
Primary: 30C15
Secondary: 30D10, 30D20  