對所有凸函數 ，則下列不等式恆成立
                 .				(1.1)
即稱為Hadamard不等式。
    我們注意到J. Hadamard不是第一個發現此不等式。正如D.S. Mitrinović和 I.B. Lačković所指出，C.Hermite比J. Hadamard早就在10年前於1883年就發現此不等式。
  Hudzik 和 Mailgranda 研究另一型態的s-凸函數，並稱為第二類s-凸函數，這種類型的函數定義如下：  對一些固定實數 而言，若函數 滿足對所有 和 [0,1]，下列不等式恆成立：
				 
則此函數稱為第二類s-凸函數，記作  。
  當 時，可輕易發現 s-凸函數變成定義域在 的一般凸函數。
  由Pearce and Pečarić和Kirmaci et al.所證出的定理是計算(1.1)式的中間項和右項的差。而這篇論文的主要研究目的則是探討(1.1)式的中間項和左項的差。

The following inequalities
                .				(1.1)
which hold for all convex mappings   are known in the literature as Hadamard’s inequality. We note that J. Hadamard was not the first who discovered them. As is pointed out by D.S. Mitrinović and I.B. Lačković, the inequalities (1.1) are due to C.Hermite who obtained them in 1883, ten years before J. Hadamard.
  Hudzik and Mailgranda considered, among others, the class of functions which are s-convex in the second sense. This class is defined in the following way: a function   is said to be s-convex in the second sense if
				 
holds for all  ,  [0,1] and for some fixed  . The class of s-convex functions in the second sense is denoted by  .
  It is easily seen that for  , s-convexity reduces to the ordinary convexity of functions defined on  .
  The theorems which were proved by Pearce and Pečarić and Kirmaci et al. are estimating the difference between the middle and right terms in (1.1). The aim of this paper is estimating the difference between the middle and left terms in (1.1).

1.Introduction……………………1
  Theorem A.………………………2
  Theorem B.………………………2
  Theorem C.………………………2
2. Main Results.…………………3
  Lemma 1.…………………………3
  Theorem 1.………………………4
  Remark 1..………………………5
  Theorem 2.………………………6
  Theorem 3.………………………8
  Corollary 1……………………10
  Corollary 2……………………10
  Corollary 3……………………11
  Remark 2.………………………12
References.………………………13
1.導論.……………………………15
  定理A..…………………………16
  定理B..…………………………16
  定理C..…………………………16
2.主要結論.………………………17
  引理1..…………………………17
  定理1……………………………18
  備註1..…………………………19
  定理2……………………………20
  定理3……………………………21
  推論1……………………………24
  推論2……………………………24
  推論3……………………………25
  備註2..…………………………26
參考文獻.…………………………26

[1]S.S. Dragomir, S. Fitzpatrick, The Hadamard’s inequality for s-convex functions in the second sense, Demonstratio Math .32 (4) (1999) 687-696.
[2]S.S. Dragomir, R.P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and trapezoidal formula, Appl. Math. Lett. 5 (1998) 91-95.
[3]H Hudzik, L. Maligranda, Some remarks on s-convex functions, Aequationes Math. 48 (1994) 100-111.
[4]U.S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput. 147 (2004) 91-95.
[5]U.S. Kirmaci, M. Klaričić Bakula, M.E. Özdemir, J. Pečarić, Hadamard-type inequalities for s-convex functions, Appl. Math. Comput. 193 (2007) 26-35.
[6]D.S. Mitrinović and I.B. Lačković, Hermite and convexity, Aequationes Math. 28 (1985) 225-232.
[7]D.S. Mitrinović, J.E. Pečarić, A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993, p.106, 10, 15.
[8]M.E. Özdemir, U.S. Kirmaci, Two new theorems on mappings uniformly continuous and convex with applications to quadrature rules and means, Appl. Math. Comput. 143 (2003) 269-274.
[9]M.E. Özdemir, A theorem on mappings with bounded derivatives with applications to quadrature rules and means, Appl. Math. Comput. 138 (2003) 425-434.
[10]B.G. Pachpatte, On some inequalities for convex functions, RGMIA Res. Rep. Coll. 6 (E) (2003). http://rgmia.vu.edu.au/v6(E).html.
[11]B.G. Pachpatte, Inequalities for Differentiable and Integral Equations, Academic Press Inc., 1997.
[12]J.E. Pečarić, F. Proschan, Y.L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press Inc., 1992, p.137.
[13]C.E.M. Pearce, J. Pečarić, Inequalities for differentiable mappings with application to special means and quardrature formulae, Appl. Math. Lett. 13 (2000) 51-55.