數個函數乘積之積分不等式之研究

INTEGRAL INEQUALITIES INVOLVING THE PRODUCT OF SEVERAL FUNCTIONS

許凱程(Kai-Chen Hsu)

指導教授 ： 楊國勝

淡江大學/理學院/數學學系碩士班/碩士(2007年)

https://doi.org/10.6846/TKU.2007.00688

摘要

在本篇論文中，建立數個函數乘積之積分不等式的一般性結果及離散模式。

一般性結果：

|product_{i=1}^{n}f_{i}(x)-[sum_{i=1}^{n}w_{i}*F_{i}*(product_{j eq i}f_{j}(x))]|
leq M*[sum_{i=1}^{n}w_{i}*(integral_{a}^{b}|f_{i}^{'}(t)|dt)*|product_{j eq i}f_{j}(x)|] (1)

離散模式：

|product_{i=0}^{m}u_{i,j}-sum_{i=0}^{m}gamma _{i}*(product_{l eq i}u_{l,j})*U_{i}|
leq M*{sum_{i=0}^{m}[gamma_{i}*|product_{l eq i}u_{l,j}|*(sum_{j=0}^{n-1}|Delta u_{i,j}|)]} (2)

上式(1)與(2)用以估計數個函數乘積及離散模式的偏差。

關鍵字

積分不等式 ； 數個函數乘積 ； 離散模式 ； 等式

並列摘要

We establish the general results of integral inequalities involving the product of several functions and their derivatives. The discrete analogues of the main results are also given.

The product of several functions:

|product_{i=1}^{n}f_{i}(x)-[sum_{i=1}^{n}w_{i}*F_{i}*(product_{j eq i}f_{j}(x))]|
leq M*[sum_{i=1}^{n}w_{i}*(integral_{a}^{b}|f_{i}^{'}(t)|dt)*|product_{j eq i}f_{j}(x)|] (1)

The discrete analogues:

|product_{i=0}^{m}u_{i,j}-sum_{i=0}^{m}gamma _{i}*(product_{l eq i}u_{l,j})*U_{i}|
leq M*{sum_{i=0}^{m}[gamma_{i}*|product_{l eq i}u_{l,j}|*(sum_{j=0}^{n-1}|Delta u_{i,j}|)]} (2)

The above inequalities (1) and (2) can be used to estimate the deviation of the product of several functions. The discrete versions of the main results are also given.

並列關鍵字

Integral inequalities ； Product of several functions ； Discrete analogues ； Identities

參考文獻

[5] B.G.PACHPATTE, A note on integral inequalities involving the product of two functions, J. of Ineq. In Pure and Appl. Math, Vol. , Issue 2, Article 78, 2006.

[6] B.G.PACHPATTE, A note on Ostrowski type inequalities, Demonstrstio Math.,35(2002),27-30.

[7] B.G.PACHPATTE, Mathematical Inequalities, North-Holland Mathematical Library, Vol. 67, Elsevier Science, 2005.

[1] E.F. BECKENBACH AND R. BELLMAN, Inequalities, Springer-Verlag, Berlin-New York, 1970.

[2] G.H. HARDY, J.E. LITTLEWOOD AND G. POLYA, Inequalities, Cambridge University Press, 1934.