Entire functions of minimal growth with prescribed zeros - Andrusyak, I.V.; Filevych, P.V. (Carpathian Mathematical Publications, 2024)
Let l be a positive continuous increasing to +∞ function on R. For a positive non-decreasing on R function h, we found sufficient and necessary conditions under which, for an arbitrary complex sequence (ζ_n) such that ζ_n→∞ as n→∞ and n(r)l(r) for all sufficiently large r, there exists an entire function f whose zeros are the ζ_n (with multiplicities taken into account) satisfying \[M(r)=o(l^-1(n(r))n_ζ(r)h( n(r))),r otin E, r→+∞, \] where E[1,+∞) is a set of finite logarithmic measure. Here, n(r) is the counting function of the sequence (ζ_n), and M(r) is the maximum modulus of the function f.