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