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A bilogarithmic criterion for the existence of a regular minorant that does not satisfy the bang condition. (English. Russian original) Zbl 1483.30007

Math. Notes 110, No. 5, 666-678 (2021); translation from Mat. Zametki 110, No. 5, 672-687 (2021).
Summary: Problems of constructing regular majorants for sequences \(\mu=\{\mu_n\}_{n=0}^{\infty}\) of numbers \(\mu_n\ge0\) that are the Taylor coefficients of integer transcendental functions of minimal exponential type are investigated. A new criterion for the existence of regular minorants of associated sequences of the extended half-line \((0,+\infty]\) in terms of the Levinson bilogarithmic condition \(M=\{\mu_n^{-1}\}_{n=0}^{\infty}\) is obtained. The result provides a necessary and sufficient condition for the nontriviality of the important subclass defined by J. A. Siddiqi. The proofs of the main statements are based on properties of the Legendre transform.

MSC:

30B10 Power series (including lacunary series) in one complex variable
30D15 Special classes of entire functions of one complex variable and growth estimates
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