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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

fα x + ( 1 - α ) y+fβ x + ( 1 - β ) y=fγ x + ( 1 - γ ) y+fδ x + ( 1 - δ ) y,

which holds for all x,yI, where I is a non-void open real interval. Here f is considered as an unknown real function and α,β,γ,δ(0,1) are fixed real constants. The main results show that, except the trivial case {α,β}={γ,δ}, a function f is a solution if and only if either f is a constant (provided that α+βγ+δ) 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 α+β=γ+δ), 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.

39B22Functional equations for real functions