Extension of a complete monotonicity theorem with applications

TL;DR

This paper extends a complete monotonicity theorem involving functions defined by integrals with positive functions, correcting previous proof errors and applying the results to special functions like the Hurwitz zeta and hypergeometric functions.

Contribution

It generalizes a previous theorem to broader parameter sets and corrects the proof, enabling new monotonicity results for special functions.

Findings

Extended the monotonicity theorem to interval parameters.

Established new Turán-type inequalities for special functions.

Corrected the proof of a key inductive step.

Abstract

Let $F_{p}(x) =L( t^{p}f(t)) =\int_{0}^{\infty }t^{p}f(t) e^{-xt}dt$ converge on $(0,\infty)$ for $p\in \mathbb{N}_{0}=\mathbb{N}\cup{0}$, where $f(t)$ is positive on $(0,\infty)$. In a recent paper [Z.-H. Yang, A complete monotonicity theorem related to Fink's inequality with applications, \emph{J. Math. Anal. Appl.} \textbf{551} (2025), no. 1, Paper no. 129600], the author proved the sufficient conditions for the function \begin{equation*} x\mapsto \prod_{j=1}^{n}F_{p_{j}}(x) -\lambda _{n}\prod_{j=1}^{n}F_{q_{j}}(x) \end{equation*} to be completely monotonic on $(0,\infty)$ by induction, where $\boldsymbol{p}_{[n] }=(p_{1},...,p_{n}) $ and $ \boldsymbol{q}_{[n] }=(q_{1},...,q_{n}) \in \mathbb{N }_{0}^{n}$ for $n\geq 2$ satisfy $\boldsymbol{p}_{[n] }\prec \boldsymbol{q}_{[n]}$ However, the proof of the inductive step is wrong. In this paper, we prove the above result also holds for $ \boldsymbol{p}_[n]},\boldsymbol{q}_{[n] }\in \mathbb{I}^{k}$, where $\mathbb{I}\subseteq \mathbb{R}$ is an interval, which extends the above result and correct the error in the proof of the inductive step mentioned above. As applications of the extension of the known result, four complete monotonicity propositions involving the Hurwitz zeta function, alternating Hurwitz zeta function, the confluent hypergeometric function of the second and Mills ratio are established, which yield corresponding Tur\'{a}n type inequalities for these special functions.