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 $\varepsilon >0$ the relation of these functions admits the estimate $|\psi_1(z)/\psi_2(z) -1| \leqslant \varepsilon R^{-\mu(1-\delta)}$ for all $|z|\leqslant R^{1-\delta}$, provided that $R\geqslant R_0$ and $R_0 =R_0(\varepsilon, \delta)$ is sufficiently large. This result is of considerable interest in the analysis of the stability in the inverse resonance problem for the Schroedinger equation.