On a functional equation containing four weighted arithmetic means

Abstract

In this paper we solve the functional equation $$f\big(\alpha x+(1-\alpha)y\big)+f\big(\beta x+(1-\beta)y\big)= f\big(\gamma x+(1-\gamma)y\big)+f\big(\delta x+(1-\delta)y\big)$$ which holds for all $x, y\in I$, where $I\subset \mathbb{R}$ is a non-void open interval, $f\colon I\to \mathbb{R}$ is an unknown function and $\alpha,\beta,\gamma,\delta\in (0,1)$ are arbitrarily fixed.