Monotonicity of some functions involving the gamma and psi functions

Abstract:

Let $L(x):=x-\frac {\Gamma (x+t)}{\Gamma (x+s)} x^{s-t+1}$, where $\Gamma (x)$ is Euler’s gamma function. We determine conditions for the numbers $s, t$ so that the function $\Phi (x):=-\frac {\Gamma (x+s)}{\Gamma (x+t)} x^{t-s-1} L^{\prime \prime }(x)$ is strongly completely monotonic on $(0, \infty )$. Through this result we obtain some inequalities involving the ratio of gamma functions and provide some applications in the context of trigonometric sum estimation. We also give several other examples of strongly completely monotonic functions defined in terms of $\Gamma$ and $\psi :=\Gamma ^{\prime }/\Gamma$ functions. Some limiting and particular cases are also considered.