A harmonic mean inequality for the $q-$gamma and $q-$digamma functions

TL;DR

This paper establishes new inequalities involving the $q$-gamma and $q$-digamma functions, demonstrating harmonic mean bounds and extremal properties that extend previous results in the literature.

Contribution

It introduces novel harmonic mean inequalities for $q$-gamma and $q$-digamma functions, generalizing known inequalities and identifying parameter ranges for extremal behavior.

Findings

Harmonic mean of $\Gamma_q(x)$ and $\Gamma_q(1/x)$ is ≥ 1 for all $x > 0$, $q otin J$.

Existence of $p_0$ such that for $q < p_0$, $\psi_q(1)$ minimizes the harmonic mean.

For $q > p_0$, $\psi_q(1)$ maximizes the harmonic mean.

Abstract

We prove amongs others results that the harmonic mean of $\Gamma_q(x)$ and $\Gamma_q(1/x)$ is greater than or equal to $1$ for arbitrary $x > 0$ and $q\in J$ where $J$ is a subset of $[0,+\infty)$. Also, we prove that for there is $p_0\in(1,9/2)$, such that for $q\in(0,p_0)$, $\psi_q(1)$ is the minimum of the harmonic mean of $\psi_q(x)$ and $\psi_q(1/x)$ for $x > 0$ and for $q\in(p_0,+\infty)$, $\psi_q(1)$ is the maximum. Our results generalize some known inequalities due to Alzer and Gautschi.