Some sharp inequalities for the Toader-Qi mean

TL;DR

This paper investigates the Toader-Qi mean, a mean related to the modified Bessel function, establishing sharp inequalities and relations with other classical means, enriching the understanding of its properties.

Contribution

The paper introduces new sharp inequalities for the Toader-Qi mean and explores its connections with power, logarithmic, and Gauss compound means.

Findings

Established sharp inequalities involving the Toader-Qi mean

Derived a chain of inequalities connecting various means

Analyzed properties of the Toader-Qi mean in relation to Bessel functions

Abstract

The Toader-Qi mean of positive numbers $a$ and $b$ defined by \begin{equation*} TQ\left( a,b\right) =\frac{2}{\pi }\int_{0}^{\pi /2}a^{\cos ^{2}\theta }b^{\sin ^{2}\theta }d\theta \end{equation*} is related to the modified Bessel function of the first kind. In this paper, we present several properties of this mean, and establish some sharp inequalities for this mean in terms of power and logarithmic means. From these a nice chain of inequalities involving Gauss compound mean, Toader mean and Toader-Qi mean is presented.