A two-parameter class of completely monotonic functions

Abstract:

Let $b\in \mathbb {R}$, let $c>0$, let $x> 0$, and let \begin{equation*} G_{b,c}(x)=\frac {e^{-x}}{x^b}P_c(x) \quad \mbox {with} \quad P_c(x)=\sum _{k=0}^\infty \frac {x^k}{\Gamma (c+k)}. \end{equation*} We prove that $G_{b,c}$ is completely monotonic on $(0,\infty )$ if and only if $b\geq 0$ and $b+c\geq 1$. Moreover, we present various functional inequalities for $P_c$. Among others, we show that if $c\in (0,1)$, then, for $x,y>0$ we have \begin{equation*} e< \frac { P_c(1/x)^x P_c(1/y)^y }{ P_c(1/(x+y))^{x+y}}. \end{equation*} If $c>1$, then the reverse inequality holds for $x,y>0$. In both cases, the constant bound $e$ is best possible.