The estimation of the ratio of two entire functions with the same zeros in the ball

Abstract

The paper studies entire functions of finite order of growth for which a representation of the form $\psi (z)=1+O(|z{|}^{-\mu }),\mu >0,$ as $z\to \infty$, is valid on a fixed ray of the complex plane. The main result is the following. Assume that the zeros of two functions ${\psi }_1,{\psi }_2$ of this class coincide in the circle of radius $R$ with the center in zero. Then given arbitrary small $\delta \in (0,1)$ and $\epsilon >0$ the relation of these functions admits the estimate $|{\psi }_1(z)/{\psi }_2(z)-1|⩽\epsilon R^{-\mu (1-\delta )}$ for all $|z|⩽R^{1-\delta }$, provided that $R⩾R_0$ and $R_0=R_0(\epsilon ,\delta )$ is sufficiently large. This result is of considerable interest in the analysis of the stability in the inverse resonance problem for the Schroedinger equation.

in Russian