Monotonicity properties of classical functions and their q-analogues

Abstract

We prove some new results and unify old ones on the complete monotonicity of functions including the gamma and digamma functions and their q-analogues. All of these results lead to new and interesting inequalities. Of particular interest, we obtain the following results: for all \(q>0\), \(q\neq 1\), \(x>0\) and \(n\in \mathbb{N}\), we have

$$\begin{aligned}\log\big(\frac{1-q^x}{1-q}\big)- \frac14\frac{3q^x+1}{q^x-1}\log q\leq\psi_q(x) \leq\log\big(\frac{1-q^x}{1-q}\big)-\frac{1}2 \frac{q^x}{q^x-1}\log q, \q^x\big(\frac{\log q}{q^x-1}\big)^nP_{n-2}(q^x)+\frac12q^x\big(\frac{\log q}{q^x-1}\big)^{n+1}P_{n-1}(q^x)\leq(-1)^{n+1}\psi^{(n)}_q(x) \ \le q^x\big(\frac{\log q}{q^x-1}\big)^nP_{n-2}(q^x)+q^x\big(\frac{\log q}{q^x-1}\big)^{n+1}P_{n-1}(q^x).\end{aligned}$$

where \(P_n(x)\) is some polynomial of degree n to be defined later.

These inequalities are the q-analogues of the classical inequalities

$$\frac1{2x}\leq\log x-\psi(x)\leq\frac1{x},$$

and

$$\frac{(n-1)!}{x^{n}}+\frac{n!}{2x^{n+1}}\leq (-1)^{n+1}\psi^{(n)}(x)\leq\frac{(n-1)!}{x^{n}}+\frac{n!}{x^{n+1}},\quad n\geq1, \ x>0.$$