A Bilogarithmic Criterion for the Existence of a Regular Minorant that Does Not Satisfy the Bang Condition

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.