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

Keywords: entire functions , perturbations , Zeros