the following double inequality f((a+b)/2)≤1/(b-a) ∫_a^b▒f(x) dx≤1/2 [f(a)+f(b)] (1.1) holds for all convex function on [a, b] is known in the literature as Hermite-Hadamard (or Hadamard ) inequality. The main purpose of this paper is to establish some generalizations and refinements of the inequality (1.1).