Abstract
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.
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Notes
If \(M(0)\ne0\), then the function \(M (x)/x\) decreases for \(x>0\), because
$$\frac{M(x)}{x}=\frac{M(x)-M(0)}{x-0}+\frac{M(0)}{x}\downarrow$$for \(0<x\uparrow\).
Formulated as a conjecture in the referee’s report.
The minorant \(\{M_n^{*}\}\) constructed in [6] only has properties a), c) from the definition of regularity.
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Translated from Matematicheskie Zametki, 2021, Vol. 110, pp. 672–687 https://doi.org/10.4213/mzm13261.
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Gaisin, R.A. A Bilogarithmic Criterion for the Existence of a Regular Minorant that Does Not Satisfy the Bang Condition. Math Notes 110, 666–678 (2021). https://doi.org/10.1134/S0001434621110031
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DOI: https://doi.org/10.1134/S0001434621110031