Abstract:
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
$\Lambda=\{\lambda_k\}_{k=1}^\infty$.
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 $\lambda_k$.
We also obtain an analog of Krein's theorem
concerning the functions whose inverse
can be expanded in the corresponding series
of simple fractions.
Keywords:entire function of exponential type, completely regular growth, series of exponentials, Leont'ev problem, function with simple real zeros.
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