Some Bernstein functions and integral representations concerning harmonic and geometric means

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 $t\in (-\min \{x,y\},\mathrm{\infty })$ and establish integral representations of the means $H_{x,y}(t)$ and $G_{x,y}(t)$.