Bounds and monotonicity results for means involving the $q$-polygamma functions

Abstract

Let $\psi_{q,n}=\left(-1\right) ^{n-1}\psi_{q}^{\left(n\right) }$ for $
n\in \mathbb{N}\cup \left\{0\right\} $ and $q\in \left(0,1\right) $, where
$\psi_{q}^{\left(n\right) }$ is the $q$-polygamma functions. In this paper,
by means of the monotonicity of means and two classes of completely
monotonicity functions, we establish lower and upper bounds for the means $
I_{\psi_{q,n},w}\left(a,b\right) $ defined, for $b>a>0$ and $w$ being
positive and integrable on $\left[a,b\right] $, by
\begin{equation*}
I_{\psi _{q,n},w}\left( a,b\right) =\psi _{q,n}^{-1}\left( \frac{
\int_{a}^{b}w\left( x\right) \psi _{q,n}\left( x\right) dx}{
\int_{a}^{b}w\left( x\right) dx}\right) ;
\end{equation*}
and prove that the sequence $\left\{ I_{\psi _{q,n},w}\left( a,b\right)
\right\} _{n\geq 0}$ is decreasing with
\begin{equation*}
\lim_{n\rightarrow \infty }I_{\psi _{q,n},w}\left( a,b\right) =a.
\end{equation*}%
Moreover, we show that, for $a,b\in \mathbb{R}$ with $a\neq b$, the function
\begin{equation*}
x\mapsto \psi _{q,n}^{-1}\left( \frac{\int_{a}^{b}w\left( t\right) \psi
_{q,n}\left( x+t\right) dt}{\int_{a}^{b}w\left( t\right) dt}\right) -x
\end{equation*}
is increasing from $\left( -\min \left\{ a,b\right\} ,\infty \right) $ onto $\left( \min \left\{ a,b\right\} ,\beta _{2}\right) $, where
\begin{equation*}
\beta _{2}=\log _{q}\left( \frac{\int_{a}^{b}w\left( t\right) q^{t}dt}{
\int_{a}^{b}w\left( t\right) dt}\right) .
\end{equation*}%
These generalize some known results.