Rocky Mountain Journal of Mathematics

Conservation of the number of zeros of entire functions inside and outside a circle under perturbations

Michael Gil’

Abstract

Let $f$ and $f ̃$ be entire functions of order less than two, and $Ω = { z ∈ ℂ : | z | = 1 }$. Let $i i n ( f )$ and $i o u t ( f )$ denote the numbers of the zeros of $f$ taken with their multiplicities located inside and outside $Ω$, respectively. Besides, $i o u t ( f )$ can be infinite. We consider the following problem: how “close” should $f$ and $f ̃$ be in order to provide the equalities $i i n ( f ̃ ) = i i n ( f )$ and $i o u t ( f ̃ ) = i o u t ( f )$? 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.

Article information

Source
Rocky Mountain J. Math., Volume 50, Number 2 (2020), 583-588.

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

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1590739291

Digital Object Identifier
doi:10.1216/rmj.2020.50.583

Mathematical Reviews number (MathSciNet)
MR4104395

Zentralblatt MATH identifier
07210980

Citation

Gil’, Michael. Conservation of the number of zeros of entire functions inside and outside a circle under perturbations. Rocky Mountain J. Math. 50 (2020), no. 2, 583--588. doi:10.1216/rmj.2020.50.583. https://projecteuclid.org/euclid.rmjm/1590739291

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