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系統識別號 U0002-2812201114411100
中文論文名稱 在有界變分函數上有關Ostrowski型之不等式研究
英文論文名稱 On inequality of Ostrowski's type for mapping of bounded variation
校院名稱 淡江大學
系所名稱(中) 數學學系博士班
系所名稱(英) Department of Mathematics
學年度 100
學期 1
出版年 101
研究生中文姓名 周義銘
研究生英文姓名 Yi-Ming Chou
學號 895190063
學位類別 博士
語文別 中文
第二語文別 英文
口試日期 2011-12-17
論文頁數 99頁
口試委員 指導教授-楊國勝
委員-高金美
委員-張慧京
委員-曾貴麟
委員-陳功宇
委員-胡德軍
委員-劉豐哲
委員-李武炎
中文關鍵字 有界變分  全變分  Ostrowski不等式 
英文關鍵字 bounded  total variation  Ostrowski inequality 
學科別分類
中文摘要 首先第一章,先介紹Ostrowski不等式令 f: [a,b] → R 在 [a,b] 上是一個有界變分的函數。則下列不等式

|∫_a^b▒〖f(x) dx-(b-a)f(x)〗|≤[1/2 (b-a)+|x-(a+b)/2|] V_a^b (f)

對於每一個 x 在 (a,b)上都成立,這裡的 V_a^b (f) 是 f 在 [a,b] 上的全變分。
第二章,我們介紹一些已建立有關於Ostrowski型的不等式。
第三章,我們要展示我們所建立的Ostrowski不等式。
第四章,我們要介紹一些特殊的加權的 Ostrowski不等式和一些特殊的改良的 Ostrowski 不等式。我們得到了幾個重要的不等式。像是不等式在有界變分函數之下加權的梯形積分及在有界變分函數之下 ‘‘加權的 Ostrowski’’ 不等式。
最後,我們要介紹特殊平均數應用在我們的主要結果上。
英文摘要 In this dissertation, it consists of five chapters.
In the first chapter, we introduce Ostrowski inequality for function of bounded variation. The inequality
|∫_a^b▒〖f(x) dx-(b-a)f(x)〗|≤[1/2 (b-a)+|x-(a+b)/2|] V_a^b (f)
holds for all x∈(a,b) where f: [a,b] → R is a mapping of bounded variation on [a,b] and V_a^b (f) is the total variation of f on the interval [a,b].
In the second chapter, we introduce Some established Ostrowski's type inequalities.
In the third chapter, we present some refinements of Ostrowski inequalities.
In the forth chapter, we present some particular weighted ostrowski inequality and some particular integral of improved ostrowski Inequality. We get some important results. Some inequalities like the weighted trapezoid inequality for mappings of bounded variation and the ‘weighted Ostrowski inequality for mappings of bounded variation.
Finally, we discuss Some Particular integral inequality about my main results.
論文目次 目錄
第一章 導論 ..............................................1
1.1 簡介...................................................1
1.2 Ostrowski 不等式.......................................2
第二章 一些已建立的 Ostrowski’s型不等式...................5
2.1由 Dragomir 所建立的結果................................5
2.2由Tseng, Hwang 和 Dragomir, Tseng 和 Hwang所建立的結果.9
第三章 一些加權的 Ostrowski 不等式的改良..................13
3.1在函數為有界變分下,加權之 Ostrowski’s 型不等式.......13
3.2在函數為有界變分下,改良之Ostrowski’s 型不等式........23
第四章 一些特殊的積分不等式...............................33
4.1一些特殊的加權的 Ostrowski不等式.......................33
4.2一些特殊的改良的 Ostrowski不等式.......................40
第五章 特殊平均數應用 ....................................44
5.1 一些特殊平均數........................................44
5.2一些特殊平均數的應用...................................46
參考文獻..................................................49

Contents
Chapter 1. Introduction..................................52
1.1 Introduction .........................................52
1.2 Ostrowski Inequality..................................53
Chapter 2.Some established Ostrowski's type Inequalities..55
2.1 Inequality established by Drangomir...................55
2.2 Inequality established by Tseng, Hwang and Drangomir..57
Chapter 3. Some Refinements of Ostrowski Inequality.......62
3.1 Weighted Ostrowski Inequality for mappings of bounded variation.................................................62
3.2 Improved Ostrowski Inequality for mappings of bounded variation.................................................72
Chapter 4. Some Particular integral inequality............82
4.1 Some Particular Weighted Ostrowski Inequality.........82
4.2 Some Particular Improved Ostrowski Inequality.........88
Chapter 5. Applications to special means..................92
5.1 Some special means....................................92
5.2 Some applications to special means....................93
References................................................97
參考文獻 [1] T. M. Apostol, Mathematical Analysis, Second Edition,
Addision-Wesley Publishing Company, 1975.
[2] S. S Dragomir and S. Wang, A new inequality of Ostrowski’s type in L_1 norm and applications to some special means and to some numerical quadrature rules, Tamkang J. of Math., 28(1997) , 239-244.
[3] S.S Dragomir and S Wang, Applications of Ostrowski's inequality to the estimation of error bounds for some special means and for some numerical quadrature rules, Appl. Math. Lett., 11(1) (1998) , 105-109
[4]. S. S. Dragomir and S. Wang, An new inequality of Ostrowski.s type in L_p [a,b]- norm, Indian J. Math. 40(3) (1998), 299-304.
[5] S. S. Dragomir, The Ostrowski integral inequality for mappings of bounded variation, Bull. Australian Math. Soc., 60(1999),495-508
[6] S. S. Dragomir, Ostrowski’s inequality for monotonous mappings and applications, J. KSIAM 3 (1) (1999), 127-135.
[7] S. S. Dragomir, The Ostrowski’s integral inequality for Lip-schitzian mappings and applications, Comput. Math. Appl., 38 (1999), 33-37.
[8] S. S. Dragomir, A New generalization of Ostrowski’s integral inequality for mappings whose derivatives are bounded and applications in numberical integration and for special means, Appl.Math. Lett., 13(2000) 19-25.
[9] S. S. Dragomir, P. Cerone and J. Roumeliotis, A new generalization of Ostrowski.s integral inequality for mappings whode derivatives are bounded and applications in numerical integration and for special means, Appl. Math. Lett., 13(1) (2000),19-25.
[10] S. S. Dragomir, On the Ostrowski.s integral inequality for mappings with bounded variation and applications, Math. Inequal.Apple., 4(1) (2001), 59-66.
[11] S. S. Dragomir, A generalization of Ostrowski integral inequality for mappings whose derivatives belong to L_p [a,b] and applications in numerical integration, J. Math. Anal. Appl., 255(2001),605-626.
[12] S. S. Dragomir, A generalization of Ostrowski integral inequality for mappings whose derivatives belong to L_1 [a,b] and applications in numerical integration, J. Comput. Anal. Appl.,3(4)(2001), 343-360.
[13] S. S. Dragomir, A generalization of Ostrowski integral inequality for mappings whose derivatives belong to L_∞ [a,b] and applications in numerical integration, J. KSIAM, 5(2)(2001), 117-136.
[14] D. S. Mitrinovi´c, J. E. Peµcari´c and A. M. Fink, Inequalitiesinvolving functions and their integrals and derivatives ,Kluwer Academic Publishers( Dordrecht), 1994.
[15] A. Ostrowski, üeber die Absolutabweichung einer differenzierbaren funktion von ihren integralmittelwert, Comment. Math.Helv. 10 (1938), 226-227 (German).
[16] J. Pečcarić and A. Vukelić, Milovanović-Pečarić-Fink , Inequality for difference of two integral means, Taiwanese J. Math.,10(4) (2006), 933-947.
[17] Kuei-Lin Tseng, Shiow-Ru Hwang, S.S. Dragomir, Generalizations of weighted Ostrowski type inequalities for mappings of bounded variation and their applications, Comput. Math.Appl., 55(8)(2008), 1785-1793.
[18] Kuei-Lin Tseng, Improvements of some inequalites of Ostrowskitype and their applications, Taiwanese J. Math., 12(9)(2008),2427-2441.
[19] Kuei-Lin Tseng, Shiow-Ru Hwang, Gou-Sheng Yang, Yi-Ming Chou, Improvements of the ostrowski integral inequality for mappings of bounded variation I, Appl. Math. Comp., 217(8)(2010)2348-2355
[20] Kuei-Lin Tseng, Shiow-Ru Hwang, Gou-Sheng Yang, Yi-Ming Chou, Weighted Ostrowski integral inequality for mappings of bounded variation, Taiwanese J. Math., 15(2)(2011),573-585.
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