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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 f ̃ be entire functions of order less than two, and Ω = { z : | 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.

Citation

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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

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

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Vol.50 • No. 2 • April 2020
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