Estimate of the Ratio of Two Entire Functions whose Zeros Coincide in the Disk

We study entire functions of finite growth order that admit the representation $\psi(z) = 1+ O(|z|^{-\mu})$, $\mu >0$, on a ray in the complex plane. We obtain the following result: if the zeros of two functions $\psi_1$, $\psi_2$ of such class coincide in the disk of radius $R$ centered at zero, then, for any arbitrarily small $\delta\in (0,1)$, $\varepsilon >0$, the ratio of these functions in the disk of radius $R^{1-\delta}$ admits the estimate $|\psi_1(z)/\psi_2(z) -1| \le \varepsilon R^{-\mu(1-\delta)}$ if $R\ge R_0(\varepsilon, \delta)$. The obtained results are important for stability analysis in the problem of the recovery of the potential in the Schrödinger equation on the semiaxis from the resonances of the operator.