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

Abstract

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