Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas ( IF 1.406 ) Pub Date : 2020-05-08 , DOI: 10.1007/s13398-020-00856-w
Zhen-Hang Yang, Jing-Feng Tian, Miao-Kun Wang

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.

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