  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.
