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。

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.

目錄

致謝....................................................I
摘要...................................................II
摘要(英文)..........................................III
目錄..................................................IV

Part1  中文
1.前言................................................1
2.主要結果.........................................2
3.離散模式........................................10
參考文獻...........................................19

Part2  英文
1.Introduction.....................................20
2.Main Results...................................21
3.The Discrete Analogues...................28
References........................................37

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