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

The Bhatia—Li mean B p x , y of positive numbers x and y is defined as 1 B p x , y = p B 1 / p , 1 / p ∫ 0 ∞ dt t p + x p 1 / p t p + y p 1 / p , p ∈ 0 , ∞ , where B · , · is the Beta function. This new family of means includes the famous logarithmic mean, the Gaussian arithmetic-geometric mean etc....

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Bibliographic Details
Published inRevista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Vol. 114; no. 3
Main Authors Yang, Zhen-Hang, Tian, Jing-Feng, Wang, Miao-Kun
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.07.2020
Springer Nature B.V
Subjects
Applications of Mathematics
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Original Paper
Parameters
Theoretical
Primary 33B15
Beta function
Hypergeometric series
Gamma function
R-function
Secondary 33C05
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Summary:The Bhatia—Li mean B p x , y of positive numbers x and y is defined as 1 B p x , y = p B 1 / p , 1 / p ∫ 0 ∞ dt t p + x p 1 / p t p + y p 1 / p , p ∈ 0 , ∞ , where B · , · 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 B p x , y is an increasing function of the parameter p on 0 , ∞ . In this paper, we give a positive answer to this conjecture. Moreover, the mean B p x , y is generalized to an multivariate mean and its elementary properties are investigated.
ISSN:1578-7303
1579-1505
DOI:10.1007/s13398-020-00856-w