A two-parameter class of completely monotonic functions
A two-parameter class of completely monotonic functions
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 …