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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 allBernstein functions of $t\in(-\min\{x,y\},\infty)$ and establish integralrepresentations of the means $H_{x,y}(t)$ and $G_{x,y}(t)$.

Some Bernstein functions and integral representations concerning harmonic and geometric means