Conservation of the number of zeros of entire functions inside and outside a circle under perturbations

Let f and f^~ be entire functions of order less than two, and Ω = {z ∈ C: |z| = 1}. Let i_in(f) and i_out(f) denote the numbers of the zeros of f taken with their multiplicities located inside and outside Ω, respectively. Besides, i_out(f) can be infinite. We consider the following problem: how “close” should f and f^~ be in order to provide the equalities i_in(f^~) = i_in(f) and i_out(f^~) = i_out(f)? If for f we have the lower bound on the boundary, that problem sometimes can be solved by the Rouché theorem, but the calculation of such a bound is often a hard task. We do not require the lower bounds. We restrict ourselves by functions of order no more than two. Our results are new even for polynomials.