Some Bernstein functions and integral representations concerning harmonic and geometric means

Abstract

It is general knowledge that the harmonic mean $H(x,y)=\frac{2}{\frac{1}{x}+\frac{1}{y}}$ 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 $G_{x,y}(t)$.

19 pages

References in corpus (5)

• Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions
• Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions
• The geometric mean is a Bernstein function
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Cited by in corpus (1)

• Completely monotonic degree of a function involving the tri- and tetra-gamma functions