If f : I → ℝ is convex on I, then f((a+b)/2)≤1/(b-a ) ∫_a^b▒〖f(x)dx ≤ 1/(2 ) [f(a)+f(b)] 〗 (1.1) This is the classical Hermite-Hadamard inequality If f is a convex function on I , do there exist real numbers l , L such that f((a+b)/2)≤ l ≤1/(b-a ) ∫_a^b▒〖f(x)dx ≤L ≤ 1/(2 ) [f(a)+f(b)] 〗 (1.2) The main purpose of this paper is to give more answers to the question (1.2)