若f：[a, b]→R 為凸函數，a, b∈ R，則
   f((a+b)/2)≤1/(b-a) ∫_a^bf(x)dx≤(f(a)+f(b))/2     (1.1)
恆成立，這就是有名的Hermite-Hadamard不等式，要探討的是，若f為[a, b]中的凸函數，是否能找到實數 l 及 L 使得下列不等式能成立：
 f((a+b)/2)≤ l ≤1/(b-a) ∫_a^bf(x)dx≤ L ≤(f(a)+f(b))/2
本論文研究的主要目的是要對上式提供更多答案。

If f:[a, b]→R is convex on [a, b],then
     f((a+b)/2)≤1/(b-a) ∫_a^bf(x)dx≤(f(a)+f(b))/2  (1.1)
is kmown in the literature the Hermite-Hadamard inequality. There is the question that if f is a convex function on [a, b], do there exist real number l and L such that
f((a+b)/2)≤l ≤1/(b-a) ∫_a^bf(x)dx≤ L ≤(f(a)+f(b))/2
The major goal of this study is to give some answers to the question

目錄

1.緒論………………………………………………P.1
  定理F……………………………………………P.1


2.主要結果…………………………………………P.2
   定理1……………………………………………P.2
   定理2……………………………………………P.3
   定理3……………………………………………P.5
   定理4……………………………………………P.7
   定理5……………………………………………P.9
參考文獻……………………………………………P.11

3.參考文獻

[1] S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities (RGMIA Monographs http://rgmia.vu.edu.au/monographs/hermits_hadamard.html),Victoria University, 2000.

[2] A.El.Farissi,Simple proof and refinement of Hermite-Hadamard inequality,J.Math ineg.ul4,No.3,(2010)365-369 

[3] D.S.Mitrinović and I.B.Lacković, Hermite and convexity,Aequationcs Math,28(1985),229-232 

[4] C.Niculescu and L.-E. Persson,Old and new on the Hermite-Hadamard inequality,Real Analysis Exchange,(2004).