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Zbl 1130.39019
Varga, Adrienn
On a functional equation containing four weighted arithmetic means. (English)
[J] Banach J. Math. Anal. 2, No. 1, 21-32, electronic only (2008). ISSN 1735-8787/e

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.
[Zsolt Páles (Debrecen)]

MSC 2000:: *39B22 Functional equations for real functions

Keywords: $p$-Wright affine function; Jensen affine function

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Zentralblatt MATH Berlin [Germany]