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′).

中文翻译：

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l-Index的有界性和整个函数的完全正则增长

我们研究了阶 𝜌 完全规则增长的整函数类与具有有界 l-index 的整函数类之间的关系，其中 l(z) = |z|𝜌−1 + 1 for |z| ≥ 1. 考虑了这些函数在微分方程解析理论中的可能应用。我们针对具有属于这些类的差异的给定属性的函数的存在性提出了三个新问题。对于第四个问题，我们得到了肯定的答案，即，我们提出了一个无限乘积是阶 𝜌 完全规则增长的完整函数的充分条件，该函数具有无界 l𝜌 指数，其零点不满足众所周知的莱文条件（ C) 和 (C')。