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
且
 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、S&aacute;ndor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、S&aacute;ndor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。

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, S&aacute;ndor 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, S&aacute;ndor 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 Fej&eacute;r’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.

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

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