On the general solution of a quartic functional equation. (English) Zbl 1048.39017

Using a classical result of M. Hosszú and applying elegant and elementary arguments the authors find the general solution of the functional equation: \[ f(x + 2y) + f(x-2y) + 6 f(x) = 4 (f (x+y) + f(x-y) + 6 f(y)), \] i.e., \( f(x) = A^4 (x)\) which is the diagonal of a 4-additive symmetric function from \(R^4\) into \(R\). By means of some results due to L. Székelyhidi, solutions of the above equation are also found in certain commutative groups.


39B22 Functional equations for real functions
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