淡江大學覺生紀念圖書館 (TKU Library)
進階搜尋


下載電子全文限經由淡江IP使用) 
系統識別號 U0002-1806201010244400
中文論文名稱 關於阿達瑪型不等式之研究及其應用
英文論文名稱 On Some Inequalities of Hadamard’s Type and Applications
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 98
學期 2
出版年 99
研究生中文姓名 許凱程
研究生英文姓名 Kai-Chen Hsu
學號 896190013
學位類別 博士
語文別 中文
第二語文別 英文
口試日期 2010-06-08
論文頁數 122頁
口試委員 指導教授-楊國勝
委員-王懷權
委員-陳功宇
委員-劉豐哲
委員-胡德軍
委員-高金美
委員-張慧京
委員-曾貴麟
委員-王忠信
中文關鍵字 赫米提-阿達瑪不等式  費伊爾不等式  凸向函數  梯形公式 
英文關鍵字 Hermite-Hadamard’s inequality  Fejer’s inequality  Convex function  Trapezoidal formula 
學科別分類
中文摘要 本篇論文共分為五章。第一章中,我們探討赫米提-阿達瑪(Hermite-Hadamard)與費伊爾(Fejér)所提出的不等式如下,令f:[a,b]->R為凸函數,g:[a,b]->R為非負可積分函數且對稱於x=(a+b)/2,則
f((a+b)/2)<=1/(b-a)*int{f(x),x=a..b}<=(f(a)+f(b))/2

f((a+b)/2)*int{g(x),x=a..b}<=int{f(x)g(x),x=a..b}<=(f(a)+f(b))/2)*int{g(x),x=a..b}
之後談論一些本論文中所引用的有關阿達瑪與費伊爾不等式的改善、推廣以及應用的結果。第二章中,我們首先介紹簡笙(Jensen)不等式,接著談論Dragomir、Hong、Milosević、Sándor與Yang所發表有關阿達瑪不等式的改善、推廣的結果,並提供詳細的證明。
在第三章中,我們將對Dragomir、Hong、Milosević、Sándor與Yang所發表的結果做進一步的推廣及改善。在第四章中,我們建立有關一凸向、可微分函數且函數求導函數加絕對值依然是凸向的不等式,此不等式與費伊爾右不等式有關,並且為Dragomir與Agarwal的推廣。
最後在第五章,我們將談論在第三章與第四章結果的應用,分別為特殊平均數、隨機變數與加權梯形公式。
英文摘要 In this dissertation, it consists of five chapters. In the first chapter, we introduce Hermite-Hadamard and Fejér inequality. The inequalities are
f((a+b)/2)<=1/(b-a)*int{f(x),x=a..b}<=(f(a)+f(b))/2
and
f((a+b)/2)*int{g(x),x=a..b}<=int{f(x)g(x),x=a..b}<=(f(a)+f(b))/2)*int{g(x),x=a..b}
where f:[a,b]->R is a convex function and g:[a,b]->R is nonnegative integral function such that g is symmetric to .In the second chapter, there is an introduction of documenting famous Jensen’s inequality and the refinements as well as the generalizations of the Hermite-Hadamard’s inequality which was found by Dragomir, Hong, Milosević, Sándor and Yang, respectively. Furthermore, we give some examples of their proof.
In the third chapter, we establish some inequalities that are related to the refinements of the Hadamard’s inequality base on Dragomir, Hong, Milosević, Sándor and Yang’s results. In the forth chapter, we establish some inequalities for differentiable convex mappings whose derivatives in absolute value are convex. This results are connected with Fejér’s inequality holding for convex mappings which are generalizations of Dragomir and Agarwal’s results.
Finally, we discuss its applications to some special means, the weighted trapezoidal formula, r-moment, and the expectation of a symmetric and continuous random variable.
論文目次 第一章 緒論……………………………….................................................1
第一節 阿達瑪(Hadamard)不等式的簡介……………………….........1
第二節 阿達瑪不等式的推廣………………………………………......6
第二章 有關阿達瑪不等式之改善………………………………….……..8
第一節 簡笙(Jensen)不等式……………………………………….........8
第二節 阿達瑪不等式之Dragomir細分………………………..….….10
第三節 阿達瑪不等式之Yang與Hong細分…………………..…..…16
第四節 阿達瑪不等式之Dragomir、Milosević與Sándor細分…..….19
第三章 有關阿達瑪不等式之探究....….………………………….…......26
第一節 Yang與Hong不等式進一步細分.……………………….…..26
第二節 Dragomir、Milosević與Sándor不等式進一步細分…….......33
第四章 費伊爾(Fejér)不等式與加權梯形公式之探究............43
第一節 加權梯形公式在費伊爾不等式型態的推廣………….….......43
第二節 加權梯形公式在費伊爾不等式型態的應用….………...........45
第五章 有關阿達瑪不等式之應用………………….........................…..49
第一節 阿達瑪型不等式在特殊平均數上的應用………………...…..49
第二節 不等式與隨機變數.....................................................................53
第三節 加權梯形公式之應用…..………………………………...…....54
參考文獻….………………………………………………………....…..….57

Contents
Chapter 1. Introduction……………………...........................................................62
1.1 Hadamard Inequality……………………………………………..…........…62
1.2 The Generalization of Hadamard Inequality……………………………......68
Chapter 2. Some Refinements of Hadamard Inequality…………………...……...69
2.1 Jensen Inequality……………………………………………...……….........69
2.2 Dragomir’s Refinements of Hadamard inequality…………………..........…71
2.3 Yang and Hong’s Refinement of Hadamard inequality……….………....….77
2.4 Dragomir, Milosević and Sándor Refinements of Hadamard inequality…....80
Chapter 3. On Some Hadamard’s Type Inequalities………………………..…......87
3.1 Further Refinements of Yang and Hong’s Inequalities………………….......87
3.2 Further Refinements of Dragomir, Milosević and Sándor’s Inequalities.......94
Chapter 4. Some Inequalities for Fejér Inequality and Trapezoidal Formula........104
4.1 The Generalization of Weighted Trapezoidal Formula on Fejér Type Inequality……………………………………………………………….......104
4.2 Applications to Weighted Trapezoidal Formula on Fejér Type Inequality....106
Chapter 5. Applications of Hadamard Inequality...................................................110
5.1 Applications to Special Means…………………………………………......110
5.2 Some Inequalities for Random Variables......................................................114
5.3 Applications to Weighted Trapezoid Formula………………….……..........115
References……………………………………………....……………………..........118
參考文獻 [1] H. Alzer. A note on Hadamard’s inequalities, C. R. Math. Rep. Acad Sci. Canada, 11(1989), 255-258.
[2] J. L. Brenner and H. Alzer, Integral Inequalities for Concave Functions with Applications to special functions, Proc. Roy. Soc. 2 Edinburgh A, 118 (1991), 173-192.
[3] N. S. Barnett, S. S. Dragomir and C. E. M. Pearce, A quasi-trapezoid inequality for double integrals, Anziam J. 44(2003) 355-364.
[4] P. S. Bullen, D. S. Mitrinovic and P. M. Vasic (Eds), Means and Their Inequalities, D. Reidel Publishing Company, 1988.
[5] P. Cerone, S. S. Dragomir and C. E. M. Pearce, A generalized trapezoid inequality for functions of bounded variation, Turkish J. Math. 24 (2000)147-163.
[6] S. S. Dragomir, A mapping in connection to Hadamard’s inequalities, Anz Oster Akad Wiss Math-Naturwiss Klasse 128(1991)17-20.
[7] S. S. Dragomir, A Refinement of Hadamard’s Inequalities for Isotonic Linear Functionals, Tamkang. J. Math.
[8] S. S. Dragomir, Further Properties of Some Mapping Associated with Hermite-Hadamard’s Inequalities, Tamkang. J. Math., 34(1) (2003), 45-57.
[9] S.S. Dragomir, On Hadamard’s inequalities for convex functions, Mat. Balkanica 6(1992), 215-222. MR: 934: 26033.
[10] S. S. Dragomir, On the Hadamard’s Inequalities for Convex on the Co-ordinates in a Rectangle form the Plane, Taiwanese J. Math., 5(4)(2001), 775-788.
[11] S.S. Dragomir, On Hadamard’s inequality for the convex mappings defined on a ball in the space and applications, Math. Ineq. & Appl., 3 (2) (2000), 177-187.
[12] S.S. Dragomir, On Hadamard’s inequality on a disk, Journal of Ineq. in Pure & Appl. Math., 1 (2000), No. 1, Article 2, http://jipam.vu.edu.au/
[13] S. S. Dragomir, Some remarks on Hadamard’s inequalities for convex functions, Extracta Math., 9 (2) (1994), 88-94.
[14] S. S. Dragomir, Two Mappings in Connection to Hadamard’s Inequalities, J. Math. Anal. Appl., 167 (1992),49-56.
[15] S. S. Dragomir, Two refinements of Hadamard’s inequalities, Zb.-Rad. (Kragujevac), (1990), No. 11, 23–26.
[16] S. S. Dragomir and R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and trapezoidal formula, Appl. Math. Lett. 11(5), 91-95, (1998).
[17] S.S. Dragomir and R.P. Agarwal, Two new mappings associated with Hadamard’s inequalities for convex functions, Appl. Math. Lett., 11 (1998), No. 3, 33-38.
[18] S.S. Dragomir and C. Bus. E, Refinements of Hadamard’s inequality for multiple integrals, Utilitas Math (Canada), 47 (1995), 193-195.
[19] S. S. Dragomir and C. E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, Victoria University of Technology, Melbourne.
[20] S. S. Dragomir, P. Cerone and A. Sofo, some remarks on the trapezoid rule in numerical integration, Indian J. Pure Appl. Math. 31(2000) 475-494.
[21] S. S. Dragomir, Y. J. Cho and S. S. Kim, Inequalities of Hadamard’s type for Lipschitzian Mappings and Their Applications, J. Math. Anal. Appl., 245(2000), 489-501.
[22] S. S. Dragomir, D. S. Milosević and József Sándor, On Some Refinements of Hadamard’s Inequalities and Applications, Univ. Belgrad. Publ. Elek. Fak. Sci. Math., 4(1993), 3-10.
[23] S. S. Dragomir, C. E. M. Pearce and J. E. Pečarić, On Jessen’s and related inequalities for isotonic sublinear functionals, Acta Math. Sci. (Szeged), 61 (1995), 373-382.
[24] S.S. Dragomir, J. E. Pečarić and J. Sándor, A note on the Jensen-Hadamard inequality, L. Anal Num Theor L Approx, 19(1990) 29-34.
[25] L. Fejér, Über die Fourierreihen, II, Math. Naturwiss. Anz Ungar Akad. Wiss., 24(1906), 369-390, in Hungarian.
[26] J. Hadamard, Étude sur les propriétés des fonctions entiéres en particulier d’une fonction considérée par Riemann, J. Math. Pures Appl., 58(1893), 171-215.
[27] Robert V. Hogg and Elliot A. Tanis, Probability and Statistical Inference, Macmillan College Publishing Company Inc., 1993.
[28] Paul G. Hoel, Sidney C. Port and Charles J. Stone, Introduction to Probability Theorem, Houghton Mifflin Company, Boston, Mass., 1971.
[29] D.-K. Hwang, K.-L. Tseng and G. S. Yang, Some Hadamard’s Inequalities for Coordinated Convex Functions in a Rectangle from the Plane, Taiwanese J. Math., 11(1) (2007), 63-73.
[30] S. S. Kragujevac, Two refinements of Hadamards’s inequalities, Coll. Sci. Pap. Fac. Sci. Kragujevac,11(1990) 23-26.
[31] K.-C. Lee and K.-L. Tseng, On a Weighted Generalization of Hadamard’s Inequality for G-convex Functions, Tamsui-Oxford J. Math. Sci., 16(1)(2000), 91-104.
[32] D. S. Mitrinovic I. B Lackovic, Hermite and convexity, Acquations Math. 28(1985) 225-232.
[33] H. L. Royden, Real Analysis, Prentice Hall, Upper Saddle River, New Jersey, 1988.
[34] J. Sandor, Some integral inequalities, Elem Math., 43(1988)177-180.
[35] K.-L. Tseng, On Inequalities Concern Convex Functions, Ph. D. Dissertation, Tamkang University, Taiwan, 2000.
[36] K.-L. Tang, S.-R. Hwang and S. S. Dragomir, On some New Inequalities of Hermite- Hadamard –Fejér Type Involving Convex Functions, Demonstratio Math., XL(1)(2007), 51-64.
[37] K.-L. Tseng, G.-S. Yang and S. S. Dragomir, Generalizations of Weighted Trapezoidal Inequality for Mappings of Bounded Variation and Their Applications, Math Computer Modelling, 40(2004), 77-84.
[38] K.-L. Tseng, G.-S. Yang and S. S. Dragomir, Generalizations of Weighted Trapezoidal Inequality for Monotonic Mappings and Applications, ANZIAM J. 48 (2007), 1-14.
[39] K.-L. Tseng, G.-S. Yang and K.-C. Hsu, On Some Inequalities of Hadamard’s type and applications, Taiwanese Journal of Mathematics, 13(6B) (2009), 1929-1948.
[40] K.-L. Tseng, G.-S. Yang and K.-C. Hsu, Some Inequalities for Differentiable Mapping and Applications to Fejér Inequality and Weighted Trapezoidal Formula, Taiwanese Journal of Mathematics, (to appear).
[41] G.-S. Yang and M.-C. Hong, A Note on Hadamard’s Inequality, Tamkang. J. Math., 28(1)(1997), 33-37.
[42] G.-S. Yang and K.–L. Tseng, On Certain Integral Inequalities Related to Hermite-Hadamard, J. Math. Appl., 239(1999), 180-187.
[43] G.-S. Yang and K.–L. Tseng, Inequalities of Hadamard’s Type for Lipschitzian Mappings, J. Math. Anal. Appl., 260(2001),230-238.
[44] G.-S. Yang and K.–L. Tseng, On Certain Multiple Integral Inequalities Related to Hermite-Hadamard Inequalities, Utilitas Math., 62(2002), 131-142.
[45] G.-S. Yang and K.–L. Tseng, Inequalities of Hermite-Hadamard-Type for Convex Functions and Lipschitzian, Taiwanese. J. Math., 7(3) (2003), 433-440.
[46] G.-S. Yang, D.-Y. Hwang and K.-L. Tseng, Some inequalities for differentiable convex and concave mappings, Comp. Math. Appl., 47(2004) 207-216.
[47] C.-S. Wang, On Interpolations of the Inequalities of Ky Fan and Hadamard, Ph. D. Dissertation, Tamkang University, Taiwan, 1997.
論文使用權限
  • 同意紙本無償授權給館內讀者為學術之目的重製使用,於2011-06-24公開。
  • 同意授權瀏覽/列印電子全文服務,於2011-06-24起公開。


  • 若您有任何疑問,請與我們聯絡!
    圖書館: 請來電 (02)2621-5656 轉 2281 或 來信