Estimate of the ratio of two entire functions whose zeros coincide in the disk

Abstract

We study entire functions of finite growth order that admit the representation ψ(z) = 1 + O(|z|^−μ), μ > 0, on a ray in the complex plane. We obtain the following result: if the zeros of two functions ψ₁, ψ₂ of such class coincide in the disk of radius R centered at zero, then, for any arbitrarily small δ ∈ (0, 1), ε > 0, the ratio of these functions in the disk of radius R^1−δ admits the estimate |ψ₁(z)/ψ₂(z) − 1| ≤ εR^{−μ(1−δ)} if R ≥ R₀(ε, δ). 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.