MONOTONICITY RESULTS FOR FUNCTIONS INVOLVING THE q-POLYGAMMA FUNCTIONS

Abstract

Let ${\psi }_{q,n}={(-1)}^{n-1}{\psi }_q^{(n)}$ for $n\in \mathbb{N}$, where ${\psi }_q^{(n)}$ are the $q$-polygamma functions. In this paper, by the monotonicity rules for the ratio of two power series, it is proved that, for $q\in (0,1)$ and $n\in \mathbb{N}$, the function

$$x\mapsto F_{q,n}(x;\alpha )=\frac{q^{x+\alpha }-1}{\text{ln }q}\frac{{\psi }_{q,n+1}(x)}{{\psi }_{q,n}(x)},$$

is decreasing (increasing) on $(0,\infty )$ if and only if $\alpha \le {\text{log }}_q(2^n∕(q+1))$ ($\alpha \ge 0$). The conditions for which several relevant functions are monotonic or completely monotonic on $(0,\infty )$ are obtained. Moreover, several relations involving the $q$-polygamma functions are established.