有關Opial 以及Hermite-Hadamard不等式的推廣與應用

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。

關鍵字

Opial不等式；時間尺度；赫米提-阿達瑪不等式；凸向函數

並列摘要

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.

並列關鍵字

Opial’s inequality ； Time scales ； Hermite-Hadamard’s inequality ； Convex function

參考文獻

[1] R. P. Agarwal, Sharp Opial-type inequalities involving r-derivatives and their applications, Tohoku Math. J. (Ser. 2) 47 (1995), 567--593.

[2] R. Agarwal, M. Bohner and A. Peterson, Inequalities on time scales: A survey, Math. Inequal. Appl. 4 (2001), 535--557.

[3] R. P. Agarwal and P. Y. H. Pang, Remarks on the generalizations of Opial's inequality, J. Math. Anal. Appl. 190 (1995), 559--577.

[6] N. S. Barnett, S. S. Dragomir and C. E. M. Pearce, A quasi-trapezoid inequality for double integrals, Anziam J. 44(2003) 355-364.

[7] P. R. Beesack, On an integral inequality of Z. Opial, Trans. Amer. Math. Soc. 104 (1962), 470--475.

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有關Opial 以及Hermite-Hadamard不等式的推廣與應用

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