Abstract
We study the relationship between the class of entire functions of completely regular growth of order 𝜌 and the class of entire functions with bounded l-index, where l(z) = |z|𝜌−1 + 1 for |z| ≥ 1. Possible applications of these functions in the analytic theory of differential equations are considered. We formulate three new problems on the existence of functions with given properties that belong to the differences of these classes. For the fourth problem, we obtain an affirmative answer, namely, we present sufficient conditions for an infinite product to be an entire function of completely regular growth of order 𝜌 with unbounded l𝜌-index whose zeros do not satisfy the well-known Levin conditions (C) and (C′). We also construct an entire function of completely regular growth of order 𝜌 with unbounded l𝜌-index whose zeros do not satisfy the Levin conditions (C) and (C′).
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