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系統識別號 U0002-1506201115380100
中文論文名稱 有關導數為第二類s-凸函數的Hermite-Hadamard不等式的研究
英文論文名稱 On Hermite-Hadamard Type Inequalities for Functions Whose Derivatives are s-Convex in the Second Sense
校院名稱 淡江大學
系所名稱(中) 中等學校教師在職進修數學教學碩士學位班
系所名稱(英) Executive Master's Program In Mathematics for Teachers
學年度 99
學期 2
出版年 100
研究生中文姓名 洪秋月
研究生英文姓名 Chiu-Yueh Hung
學號 798190137
學位類別 碩士
語文別 英文
第二語文別 中文
口試日期 2011-06-04
論文頁數 27頁
口試委員 指導教授-楊國勝
委員-楊國勝
委員-曾貴麟
委員-張慧京
中文關鍵字 凸函數  s-凸函數  Hadamard不等式 
英文關鍵字 convex  s-convex  Hadamard’s inequality 
學科別分類
中文摘要 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。
英文摘要 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).
論文目次 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
參考文獻 [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.
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