浅谈积分不等式的证明.doc

浅谈积分不等式的证明 摘 要 积分不等式的证明方法灵活多样，技巧性和综合性较强。每种方法有一定的特色，并且有一定的规律可循。本文综述了积分不等式的若干方法。通过对例题的分析，总结了求积分不等式的常用方法。 这篇文章主要有两部分组成，其一，利用定积分的性质，微分中值定理，积分中值定理，概率论知识，施瓦兹不等式，二重积分等内容，研究了积分不等式的证法。其二，研究了Gronwall积分不等式不同的证明方法并加以应用。更重要的是，对某些积分不等式进行推广。 [关键词]：定积分,概率论，积分不等式，泰勒公式 Abstract The proof of integral inequality is flexible，skillful and complex . Every method has its feature. However, it also has law to obey. The article explains some methods. By analysis course of some examples, I sum up some methods of proving integral inequality. The article mainly has two aspects. Firstly, the article explores ten methods of proving Integral inequality with the nature of definite integral，Mean value theorem of differential, mean value theorem of integral，Schwarz inequality, Taylor formula, probability knowledge and double integral and so on. Secondly, the article has studied the proof of Gronwall integral inequality and its application. What is more, some integral inequalities have been generalized by the article. [Keywords]:Definite Integral, Probability, Integral Inequality ,Taylor formula. 目录 引言..............................................................1 积分不等式的证明方法......................................2 1.1利用定积分性质证明积分不等式...................................2 1.2利用中值定理证明积分不等式.....................................3 1.3利用施瓦兹不等式证明积分不等式.................................4 1.4利用二重积分证明积分不等式.....................................5 1.5利用反证法证明积分不等式.......................................6 1.6利用线性变换证明积分不等式.....................................7 1.7利用泰勒公式证明积分不等式.....................................7 1.8作辅助函数利用函数单调性证明积分不等式.........................8 1.9利用概率论方法证明积分不等式...................................8 1.10利用Gurland不等式证明积分不等式..............................10 第二章 一些特殊积分不等式的证明，推广，及应用.....................12 2.1Gronwall积分不等式的证明及其应用...............................12 2.2对某个积分不等式的推广.........................................15 2.3数值积分不等式.................................................16 2.4 Steffens