Hyers-Ulam-Rassias stability of a quadratic functional equation

Abstract

In this paper we deal with the quadratic functional equationbegin{eqnarray*}& & n^2binom{n-2}{k-2}fleft(frac{x_1+cdots+x_n}{n}right)+binom{n-2}{k-1}sum_{i=1}^n f(x_i) & =& k^2sum_{1leq i_1<cdots<i_kleq n}fleft(frac{x_{i_1}+cdots+x_{i_k}}{k}right),end{eqnarray*}deriving from an inequality of T. Popoviciu for convex functions. We solve this functional equation by proving thatits solutions are the polynomials of degree at most two. Likewise, we investigate its stability in the spirit ofHyers, Ulam, and Rassias.