若f: [a,b]→R為凸函數, a,b∈R ,則f((a+b)/2) ≤ 1/(b-a)∫_a^b f(x)dx ≤ 1/(2) [f(a)+f(b)]恆成立, 這就是著名的Hermite-Hadamard不等式, 繼續探討的是,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) ≤ 1/(b-a)∫_a^b f(x)dx ≤ 1/(2) [f(a)+f(b)] 本論文研究的主要目的是要對上式提供一些答案。

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)] is known 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 numbers l and L such that f((a+b)/2) ≤ 1/(b-a)∫_a^b f(x)dx ≤ 1/(2) [f(a)+f(b)] ? The major goal of this study is to give some answers to the question.

1.緒論……………………………………………………………………………1
2.主要結果……………………………………………………………………3
3.參考文獻……………………………………………………………………21

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

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

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

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