I have not any idea, how to attack the equation $$f\left( \dfrac{2xy}{x+y}\right) +f\left( \dfrac{x+y}{2}\right) =f\left( x\right) +f\left( y\right)$$ with unknown $f:\mathbb{R} _{+}\rightarrow \mathbb{R}$.
Allowing (for a while) that $y$ can be equal to $0$, we get $$\begin{aligned}f\left( \dfrac{x}{2}\right) =f\left( x\right) \\ f\left( \dfrac{x}{2^{n}}\right) =f\left( x\right) \\ f\left( 2x\right) =f\left( x\right) \\ f\left( 2^{n}x\right) =f\left( x\right) \end{aligned}$$
This suggests that f can be constant. Otherwise, as $$\begin{aligned}HM\left( x,y\right) \cdot AM\left( x,y\right) =GM^{2}\left( x,y\right)= xy,\end{aligned} $$
one can guess that (up to a constant) $f$ can be a logarithmic function. So, $f(x) =A\ln x+B$. How to perform a rigorous proof of these ( maybe without guessing ?) How to prove, that there's not other solutions ? Thanks in advance for your help...
