The problem of Leont'ev on entire functions of completely regular growth

We consider an entire function of exponential type with all its zeros are simple and form a sequence with the index condensation zero. On the set of zeros a function of its derivative is growing quickly. Required to determine whether original function have complete regularity of growth. This problem, which arose in the theory of representation of analytic functions by exponential series was posed by A. F. Leontiev more than forty years ago and has not yet been solved. In this paper we show that the aforesaid problem a positive solution if the function is “not too small” on a straight line.