## Rocky Mountain Journal of Mathematics

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

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

#### Abstract

Let $f$ and $\stackrel{\u0303}{f}$ be entire functions of order less than two, and $\mathrm{\Omega}=\left\{z\in \u2102:\left|z\right|=1\right\}$. 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 $\stackrel{\u0303}{f}$ be in order to provide the equalities ${i}_{in}\left(\stackrel{\u0303}{f}\right)={i}_{in}\left(f\right)$ and ${i}_{out}\left(\stackrel{\u0303}{f}\right)={i}_{out}\left(f\right)$? 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.

#### 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

**Subjects**

Primary: 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10}

Secondary: 30D10: Representations of entire functions by series and integrals 30D20: Entire functions, general theory

**Keywords**

entire functions zeros perturbations

#### 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