Let and be entire functions of order less than two, and . Let and denote the numbers of the zeros of taken with their multiplicities located inside and outside , respectively. Besides, can be infinite. We consider the following problem: how “close” should and be in order to provide the equalities and ? If for 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.
"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