On a functional equation containing four weighted arithmetic means (Q2461442)

The author offers a complete discussion and solution of 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 \(I\) is a non-void open real interval. Here \(f\) is considered as an unknown real function and \(\alpha,\beta,\gamma,\delta\in(0,1)\) are fixed real constants. The main results show that, except the trivial case \(\{\alpha,\beta\}=\{\gamma,\delta\}\), a function \(f\) is a solution if and only if either \(f\) is a constant (provided that \(\alpha+\beta\neq\gamma+\delta\)) or \(f\) is the sum of a Jensen affine function (which is the sum of a constant and an additive function) and a quadratic function (provided that \(\alpha+\beta=\gamma+\delta\)), where the quadratic function satisfies a certain homogeneity condition depending on the constants. Thus the quadratic part of the solution can only be nontrivial if the constants satisfy a further nontrivial algebraic property.