本篇論文共分為四章。第一章中，我們探討Opial所提出的不等式。因為Opial不等式有連續型跟離散型的情形，所以我們希望透過時間尺度(time scales)的觀念將二者結合。
第二章中我們提出了一些Opial不等式在時間尺度上的一些推廣。
第三章中，我們探討赫米提-阿達瑪(Hermite-Hadamard)與費伊爾(Fejer)所提出的不等式。之後並談論一些有關阿達瑪與費伊爾不等式的改善。
最後在第四章中，我們將談論在第三章結果的應用，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 four chapters. In the first chapter, we introduce Opial’s inequality.Since there are continuous type and discrete type of Opial inequality, so we hope  to combine of both by concept of time scales.
In the second chapter, we have some improvement of Opial’s inequalities on time scales.
In the third chapter, we introduce Hermite-Hadamard and Fejer inequality.
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.

目錄
第一章 簡介----------------------------------------1
1.1 Opial不等式的介紹------------------------------1
1.2有關連續型Opial不等式的推廣-----------------------1
1.3有關離散型Opial不等式的推廣-----------------------3
1.4有關時間尺度(Time scales)的簡介------------------4
1.5一些在時間尺度上Opial不等式的推廣------------------7
第二章  一些在時間尺度上Opial不等式的改良--------------8
2.1一些在時間尺度上Opial不等式的改良------------------8
2.2一些在時間尺度上Maroni不等式的改良----------------15
第三章 阿達瑪(Hadamard)不等式的介紹-----------------21
3.1 阿達瑪(Hadamard)不等式------------------------21
3.2 一些有關於阿達瑪(Hadamard)不等式的細分-----------22
3.3一些阿達瑪不等式的推廣---------------------------23
第四章 應用---------------------------------------32
參考文獻------------------------------------------36

contents
Chapter 1 Introduction---------------------------40
1.1The Introduction of Opial's Inequalities------40
1.2 Some generalizations of Opial's Inequality in Continuous 
type---------------------------------------------40
1.3 Some generalization of Opial's Inequality in Discrete 
type---------------------------------------------42
1.4 The Introduction of Time scales--------------43
1.5 Some generalizations of Opial's inequalities on time 
scales-------------------------------------------46
Chapter 2 Some improvement of Opial's inequalities on time 
scales-------------------------------------------48
2.1 Some improvement of Opial's inequalities on time scales-----48
2.2 Some improvements of Maroni's inequalities on time scales-55
Chapter 3 Introductions of Hadamard's inequalities-62
3.1 Hadamard’s inequality--------------------------62
3.2 Some refinements of Hadamard’s inequality------63
3.3 Some generalizations of Hadamard's inequalities--------------63
Chapter 4  Applications----------------------------73
References-----------------------------------------77

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