MathDB

Let

I

be a nonempty open subinterval of the set of positive real numbers. For which even

n \in \mathbb{N}

are there injective function

f: I \to \mathbb{R}

and positive function

p: I \to \mathbb{R}

, such that for all

x_1 , \ldots , x_n \in I

f \left( \frac{1}{2} \left( \frac{x_1+\cdots+x_n}{n}+\sqrt[n]{x_1 \cdots x_n} \right) \right)=\frac{p(x_1)f(x_1)+\cdots+p(x_n)f(x_n)}{p(x_1)+\cdots+p(x_n)}

holds?

Problem Statement