Bulletin of the
Korean Mathematical Society BKMS

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

Keywords: Hyers--Ulam--Rassias stability, quadratic functional equation, Popoviciu's

MSC numbers: 39B72