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A positive answer to Bhatia—Li conjecture on the monotonicity for a new mean in its parameter
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.
更新日期:2020-05-08

 

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