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arXiv · 1301.6430

Some Bernstein functions and integral representations concerning harmonic and geometric means

Feng Qi·Xiao-Jing Zhang·Wen-Hui Li
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Abstract

It is general knowledge that the harmonic mean $H(x,y)=\frac2{\frac1x+\frac1y}$ and that the geometric mean $G(x,y)=\sqrt{xy}\,$, where $x$ and $y$ are two positive numbers. In the paper, the authors show by several approaches that the harmonic mean $H_{x,y}(t)=H(x+t,y+t)$ and the geometric mean $G_{x,y}(t)=G(x+t,y+t)$ are all Bernstein functions of $t\in(-\min\{x,y\},\infty)$ and establish integral representations of the means $H_{x,y}(t)$ and $G_{x,y}(t)$.

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Feng Qi, Xiao-Jing Zhang, Wen-Hui Li. 2013-01-28. Some Bernstein functions and integral representations concerning harmonic and geometric means. https://doi.org/10.1007/s40590-016-0085-y

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