若f:[a,b]→ℝ是一個定義在[a,b]的凸函數，則
f((a+b)/2)≤1/(b-a) ∫_a^b f(x)dx≤1/2[f(a)+f(b)]   恆成立
是著名的Hermite-Hadamard雙邊不等式。
    最近在[3]中，作者們提出了這樣的問題：若f是一個定義在[a,b]的凸函數，0<α≤1/2，則是否存在實數l和L使得下列不等式成立
 f((a+b)/2)≤l(α)≤1/(b-a) ∫_a^b f(x)dx≤L(α)≤1/2 [f(a)+f(b)]。
    本論文主要研究目的是提供上述問題的答案，並針對所找出的l和L 進行排序。

If f:[a,b]→R is convex on [a,b], then
f((a+b)/2)≤1/(b-a) ∫_a^b f(x)dx≤1/2[f(a)+f(b)]      (1.1)                                                       
is known as the Hermite-Hadamard inequality.
    Recently in [3], the authors raised such question:
If f is convex function on [a,b], 0<α≤1/2, do there exist real numbers l and L such that f((a+b)/2)≤l(α)≤1/(b-a) ∫_a^b f(x)dx≤L(α)≤1/2[f(a)+f(b)]?                 (1.2)                       
    The main purpose of this paper is to give more answers to the question, and find the relation between those l's and L's.

1.緒論...............................................1
   定理F.............................................1
2.預備定理...........................................2
3.主要結果...........................................4
   定理3.1...........................................4
   定理3.2...........................................8
   定理3.3..........................................10
   定理3.4..........................................11
參考文獻............................................13

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

[2]  A.El.Farissi,Simple proof and refinement of Hermite-Hadamard inequality,J.Math Ineq.Vol.4,No.3 (2010) 365-369

[3]  A.El.Farissi,Z.Latrench,B.Belaidi, Hadamard.Type inequalities for twice differentiable functions,RGMIA Research Report Collection,12,1(2009),Art.6.

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

[5]  C.Niculescu and L.-E. Persson,Old and New on the Hermite-Hadamard inequality,Real Analysis Exchange,2004