On the equivalence of functional equations involving means and solution to a problem of Daróczy

Summary.

In this work we prove the equivalence of the two functional equations

$$f({\mathcal{A}}(x, y; p)) + f ({\mathcal{H}}(x, y; 1 - p)) = f(x) + f(y)\quad x, y \in I,$$

and

$$2f(\mathcal{G}(x, y)) = f(x) + f(y)\quad x, y \in I.$$

Here I is a nonempty open interval of the positive real line, and ${\mathcal{A}}(x, y; p), {\mathcal{H}}(x, y; 1 - p), {\mathcal{G}}(x, y)$ are the weighted arithmetic mean with weight p, the weighted harmonic mean with weight 1 − p, and the geometric mean, respectively.