Abstract. In this paper we prove a generalization of the classical Hermite--Hadamard inequality for convex (concave) functions extending it to $n$ different nodes, without using any further restrictions on the function. We use the concept of iterated integrals of $f$ to effect our purpose. Moreover we apply the main result to the function family $f(u) = u^\alpha(\alpha\ge 0)$ and useful identities for sums and products are given.

AMS Subject Classification (1991): 26A51

Received September 12, 2005, and in final form February 1, 2008. (Registered under 6005/2009.)