A positive answer to Bhatia—Li conjecture on the monotonicity for a new mean in its parameter

Abstract

The Bhatia—Li mean $\mathcal {B}_{p}\left( x,y\right)$ of positive numbers x and y is defined as

$$\begin{aligned} \frac{1}{\mathcal {B}_{p}\left( x,y\right) }=\frac{p}{B\left( 1/p,1/p\right) } \int _{0}^{\infty }\frac{dt}{\left( t^{p}+x^{p}\right) ^{1/p}\left( t^{p}+y^{p}\right) ^{1/p}}\text {, }\ p\in \left( 0,\infty \right) , \end{aligned}$$

where $B\left( \cdot ,\cdot \right)$ is the Beta function. This new family of means includes the famous logarithmic mean, the Gaussian arithmetic-geometric mean etc. In 2012, Bhatia and Li conjectured that $\mathcal {B}_{p}\left( x,y\right)$ is an increasing function of the parameter p on $\left[ 0,\infty \right]$. In this paper, we give a positive answer to this conjecture. Moreover, the mean $\mathcal {B} _{p}\left( x,y\right)$ is generalized to an multivariate mean and its elementary properties are investigated.