Rocky Mountain Journal of Mathematics

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

Michael Gil’

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

entire functions zeros perturbations


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.

Export citation


  • P. Borwein and T. Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics 161, Springer, 1995.
  • J. G. Clunie and A. Edrei, “Zeros of successive derivatives of analytic functions having a single essential singularity, II”, J. Analyse Math. 56 (1991), 141–185.
  • A. Edrei, E. B. Saff, and R. S. Varga, Zeros of sections of power series, Lecture Notes in Mathematics 1002, Springer, 1983.
  • S. Edwards and S. Hellerstein, “Non-real zeros of derivatives of real entire functions and the Pólya–Wiman conjectures”, Complex Var. Theory Appl. 47:1 (2002), 25–57.
  • M. I. Gil', “Perturbations of zeros of a class of entire functions”, Complex Variables Theory Appl. 42:2 (2000), 97–106.
  • M. Gil', Localization and perturbation of zeros of entire functions, Lecture Notes in Pure and Applied Mathematics 258, CRC Press, Boca Raton, FL, 2010.
  • M. Gil', “Conservation of the number of eigenvalues of finite dimensional and compact operators inside and outside circle”, Funct. Anal. Approx. Comput. 10:2 (2018), 47–54.
  • M. H. Gulzar, “On the location of zeros of a polynomial”, Anal. Theory Appl. 28:3 (2012), 242–247.
  • G. V. Milovanović, D. S. Mitrinović, and T. M. Rassias, Topics in polynomials: extremal problems, inequalities, zeros, World Scientific, River Edge, NJ, 1994.
  • P. C. Rosenbloom, “Perturbation of the zeros of analytic functions, I”, J. Approximation Theory 2 (1969), 111–126.
  • S. Saks and A. Zygmund, Analytic functions, 2nd ed., Monografie Matematyczne 28, Państwowe Wydawnietwo Naukowe, Warsaw, 1965.
  • D. M. Simeunović, “On the location of zeros of some polynomials”, Math. Morav. 16:2 (2012), 59–61.
  • H. A. Soleiman Mezerji and M. Bidkham, “Cauchy type results concerning location of zeros of polynomials”, Acta Math. Univ. Comenian. $($N.S.$)$ 83:2 (2014), 267–279.