A complete monotonicity theorem related to Fink's inequality with applications

Let F be a completely monotonic function on $(0,\infty )$ and $F_n\left(x\right)={(-1)}^n{F}^{\left(n\right)}\left(x\right)$ for $n\in N_0$. Fink in 1982 proved the inequality$\prod _{i=1}^k{F}_{p_i}\left(x\right)\le \prod _{i=1}^k{F}_{q_i}\left(x\right)$for $x>0$, where $p_{\left[k\right]}=(p_1,...,p_k)$ and $q_{\left[k\right]}=(q_1,...,q_k)\in N_0^k$ for $k\ge 2$ satisfy $p_{\left[k\right]}\prec q_{\left[k\right]}$. Inspired by Fink's inequality, we further give the sufficient conditions for the function$x\mapsto \prod _{i=1}^k{F}_{p_i}\left(x\right)-{\lambda }_k\prod _{i=1}^k{F}_{q_i}\left(x\right)$to be completely monotonic on $(0,\infty )$. Applying this result, we reprove the complete monotonicity involving polygamma functions, and present some new consequences involving Nielsen's beta function and confluent hypergeometric function of the second kind.