An olympiad problem

Question

I was reading a book about olympiad problems and i encountered this exercise.

Let $f:(0,\infty) \longrightarrow \mathbb{R} $ such that $\frac{f(x)+f(y)}{2} =f(\sqrt{xy})$.

Prove that $\forall x,y,z>0$ $$\frac{f(x)+f(y)+f(z)}{3} =f(\sqrt[3]{xyz})$$

I only managed to prove it for an even number of values but i find difficulties to prove it for odd number of values.

Also we can easily see that a function with this property is $f(x)=lnx$.

Can someone give me a hint??

Thank you in advance!

Answer 1

$\frac{f(x) + f(y) + f(z) + f(\sqrt[3]{xyz})}{4} = f(\sqrt[3]{xyz})$ (just apply first formula two times)

Answer 2

Another example function is the constant function