Some Bernstein functions and integral representations concerning harmonic and geometric means

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 $G_{x,y}(t)$.