Let $f:R^{+}\to R$ and such foy any $x,y>0$ have $$f(\frac{x+y}{2})+f(\frac{2xy}{x+y})=f(x)+f(y)$$ show that $$2f(\sqrt{xy})=f(x)+f(y),\forall x,y>0$$
I find this result surprising because the arithmetic mean, the geometric mean, the harmonic mean all appear in this function,But I can't prove it,Thanks