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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.

MSC:

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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References:

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