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本科毕业设计（论文）

( 2013 届 )

题 目： 不等式的证明及其运用 专 业： 数学与应用数学 班 级： 09数学与应用数学 姓 名： 王乃泽 学 号： 09205013247 指导教师： 欧建光 职 称： 副教授 完成日期： 2013年4月20日

不等式证明及其运用

王乃泽

（温州大学瓯江学院，浙江温州，325027）

摘要：不等式证明及其应用在数学中有着不可或缺作用和地位，从初等数学到高等数学，不等式一直同我们形影不离，它的应用范围非常广泛，是数学教学容重要组成部分。在不等式的证明过程中需要用到诸多的数学思想，结合了许多重要的数学内容，本篇论文主要介绍几个著名不等式之间证明，运用，以及联系，帮助大家区分解决如何合理有效的运用这些不等式来达到自己所想要的预期效果。这几个不等式也是我们经常在学习中所要用的，具体的来说，就是通过凸函数的相关定义及其性质，进而引入Jensen不等式，由Jensen不等式推导所要的holder不等式，从holder不等式中我们看出，只要稍加变形就是大家广为熟知的柯西不等式。而柯西不等式是本篇论文讨论的重点内容，我们将着重讨论柯西不等式的几种主要表现形式及相关的证明，应用举例等等。在此之后我们还将通过柯西不等式推导著名的均值不等式，从均值不等式回到Jensen不等式的相关内容。至此，为本篇论文所论述的重要内容。 关键词：凸函数；不等式；

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Inequality proof and its application

Wangnaize

Oujiang College, Wenzhou University，Wenzhou, Zhejiang，325027

Abstract：Inequality proof and its application in mathematics has a indispensable role and status, from elementary mathematics to higher mathematics, inequality has been and we were like peas and carrots, its application range is very wide, is an important part of the capacity of mathematics teaching. In the inequality proof process need to use many mathematical thought, combined with many important mathematical content, this paper mainly introduces several famous between inequality proof, use, and contact, help you distinguish between solve how to reasonably and effectively use the inequality to achieve their desired expected effect. These a few inequality is we often in the study will use, concrete, it is through the convex function related definition and nature, and then introduce Jensen inequality, Jensen inequality is derived by the holder inequality, from holder inequality we see, as long as everyone is a widely known as the deformation of Cauchy inequality. And Cauchy inequality is discussed in this paper the key content, we will mainly discuss the Cauchy inequality several main forms and relevant proof, examples of application and so on. After that we will through the Cauchy inequality is famous mean inequality, from mean inequality back to Jensen inequality related content. So far, this paper discusses the important content. Keywords: convex functions; Inequality;

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