Relations for Imaginary Parts of Zeros of Entire Functions

Abstract

Finite order entire functions are considered. New relations for the imaginary parts of the zeros are derived. They particularly generalize the Cartwright-Levinson theorem. By virtue of these relations, under some restriction, the Hadamard theorem on the convergence exponent of the zeros is improved. 1. The main result. Consider the finite order entire function (1.1) f(λ) = ∞ ∑ k=0 akλ k (k!)γ , λ ∈ C, a0 = 1, γ > 0, with complex, in general, coefficients. Assume that (1.2) w(f) ≡ ∞ ∑ k=1 |ak| m. AMS Mathematics Subject Classification. 30D20, 30C15.