MathDB

Prove that if the function

f

is defined on the set of positive real numbers, its values are real, and

f

satisfies the equation

f\left( \frac{x+y}{2}\right) + f\left(\frac{2xy}{x+y} \right) =f(x)+f(y)

for all positive

x,y

, then

2f(\sqrt{xy})=f(x)+f(y)

for every pair

x,y

of positive numbers.

Problem Statement