On a Functional Equation Containing Weighted Arithmetic Means

Abstract

In this paper we solve the functional equation

$$ \sum\limits_{i = 1}^n {a_i f(\alpha _i x + (1 - \alpha _i )y) = 0} $$

which holds for all x, y ∈ I, where I ⊂ ℝ is a non-void open interval, f : I → ℝ is an unknown function and the weights α _i ∈ (0, 1) are arbitrarily fixed (i = 1, . . ., n). It will be proved that all solutions are generalized polynomials of degree at most n − 2. Furthermore we give a sufficient condition for the existence of nontrivial solutions.

This research has been supported by the Hungarian Scientific Research Fund (OKTA) Grants F049212, NK-68040.