On some criteria for completely regular growth of entire functions of exponential type

This paper sharpens the author’s previous results concerning the completely regular growth of an entire function of exponential type all of whose zeros are simple, forming a sequence Λ = {λ_k} _k=1 ^∞ . For a function with real zeros, we write the growth regularity conditions (on the real axis and on the entire plane) in terms of lower bounds only for the absolute value of the derivative at the points λ_k. We also obtain an analog of Krein’s theorem concerning the functions whose inverse can be expanded in the corresponding series of simple fractions.