Conservation of the number of zeros of entire functions inside and outside a circle under perturbations. (English) Zbl 1441.30010

Summary: Let \(f\) and \(\widetilde{f}\) be entire functions of order less than two, and \(\Omega = \{z \in \mathbb{C} : | z | = 1\} \). Let \(i_{\text{in}} (f)\) and \(i_{\text{out}} (f)\) denote the numbers of the zeros of \(f\) taken with their multiplicities located inside and outside \(\Omega\), respectively. Besides, \( i_{\text{out}} (f)\) can be infinite. We consider the following problem: how “close” should \(f\) and \(\widetilde{f}\) be in order to provide the equalities \(i_{\text{in}} (\widetilde{f} ) = i_{\text{in}} (f)\) and \(i_{\text{out}} (\widetilde{f} ) = i_{\text{out}} (f)\)? 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.


30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30D20 Entire functions of one complex variable (general theory)
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[1] P. Borwein and T. Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics 161, Springer, 1995. · Zbl 0840.26002
[2] 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. · Zbl 0737.30022 · doi:10.1007/BF02820463
[3] A. Edrei, E. B. Saff, and R. S. Varga, Zeros of sections of power series, Lecture Notes in Mathematics 1002, Springer, 1983. · Zbl 0507.30001
[4] 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. · Zbl 1021.30026 · doi:10.1080/02781070290002912
[5] M. I. Gil’, “Perturbations of zeros of a class of entire functions”, Complex Variables Theory Appl. 42:2 (2000), 97-106. · Zbl 1021.30024 · doi:10.1080/17476930008815275
[6] M. Gil’, Localization and perturbation of zeros of entire functions, Lecture Notes in Pure and Applied Mathematics 258, CRC Press, Boca Raton, FL, 2010. · Zbl 1195.30003
[7] 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. · Zbl 1392.15017
[8] M. H. Gulzar, “On the location of zeros of a polynomial”, Anal. Theory Appl. 28:3 (2012), 242-247. · Zbl 1274.30009
[9] G. V. Milovanović, D. S. Mitrinović, and T. M. Rassias, Topics in polynomials: extremal problems, inequalities, zeros, World Scientific, River Edge, NJ, 1994. · Zbl 0848.26001
[10] P. C. Rosenbloom, “Perturbation of the zeros of analytic functions, I”, J. Approximation Theory 2 (1969), 111-126. · Zbl 0223.46046
[11] S. Saks and A. Zygmund, Analytic functions, 2nd ed., Monografie Matematyczne 28, Państwowe Wydawnietwo Naukowe, Warsaw, 1965. · Zbl 0136.37301
[12] D. M. Simeunović, “On the location of zeros of some polynomials”, Math. Morav. 16:2 (2012), 59-61. · Zbl 1299.12002 · doi:10.5937/MatMor1202059S
[13] H.
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