關於阿達瑪型不等式之研究及其應用

On Some Inequalities of Hadamard’s Type and Applications

許凱程(Kai-Chen Hsu)

指導教授 ： 楊國勝

淡江大學/理學院/數學學系博士班/博士(2010年)

https://doi.org/10.6846/TKU.2010.00467

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
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、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。

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

並列摘要

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.

並列關鍵字

Hermite-Hadamard’s inequality ； Fejer’s inequality ； Convex function ； Trapezoidal formula

參考文獻

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