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On a functional equation containing four weighted arithmetic means. (English) Zbl 1130.39019

The author offers a complete discussion and solution of the functional equation

$f\left(\alpha x+\left(1-\alpha \right)y\right)+f\left(\beta x+\left(1-\beta \right)y\right)=f\left(\gamma x+\left(1-\gamma \right)y\right)+f\left(\delta x+\left(1-\delta \right)y\right),$

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 $\alpha ,\beta ,\gamma ,\delta \in \left(0,1\right)$ are fixed real constants. The main results show that, except the trivial case $\left\{\alpha ,\beta \right\}=\left\{\gamma ,\delta \right\}$, a function $f$ is a solution if and only if either $f$ is a constant (provided that $\alpha +\beta \ne \gamma +\delta$) 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.

##### MSC:
 39B22 Functional equations for real functions