若f：[a, b]→R 為凸函數，a,b∈R，則
f((a+b)/2) ≤ 1/(b-a) ∫ f(x) dx ≤ 1/2 [f(a)+f(b)]
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使得下列不等式成立：

f((a+b)/2) ≤ l ≤ 1/(b-a) ∫ f(x) dx ≤ L≤ 1/2 [f(a)+f(b)]

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) ∫ 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 number l and L such that

f((a+b)/2) ≤ l ≤ 1/(b-a) ∫ f(x) dx ≤ L≤ 1/2 [f(a)+f(b)]

The major goal of this study is to give some answers to the question.

1. 引言………………………………………………P.1
定理F……………………………………………P.1
目的………………………………………………P.1
2.主要結果…………………………………………P.1
定理2.1…………………………………………P.3
定理2.2…………………………………………P.6
定理2.3…………………………………………P.9
定理2.4…………………………………………P.11
定理2.5…………………………………………P.14
定理2.6…………………………………………P.17
參考文獻……………………………………………P.20

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Math. Ineq., Vol. 4, No. 3 (2010), 365-369.
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differentiable function, RGMIA Research Report Collection, 12, 1 (2009), Art.
6.
[3] J. Hadamard, Etude sur les propriétés des fonctions entières et en particulier
d’une fonction considérée par Riemann, J. Math. Pures Appl., 58 (1893),
171-215.
[4] D.S. Mitrinović and I.B. Lacković, Hermite and convexity, Aequat. Math., 28
(1985), 229-232.