摘要不等式理论是数学领域的重要理论，而均值不等式是不等式理论的核心内容，因此认真地对均值不等式进行探讨显得尤为重要。均值不等式不论在初等数学还是在高等数学都有着重要而广泛的应用，这需要对均值不等式有一个透彻的认识，因而有必要对均值不等式进行全面深入的研究。86488

     均值不等式的探讨主要内容包括了均值不等式的形式与性质、均值不等式的推广与证明、均值不等式的应用与变式探究等。本文综述了探究均值不等式的各方面所取得的主要研究成果。

     主要研究成果包括探析了均值不等式的来源，将均值不等式证明方法作了代数与几何的划分，并对比了两者各自的优劣，同时归纳了均值不等式在形式与性质上的推广。运用均值不等式时容易产生各种错误，本文结合例题作了细致的论述。

    目前该研究存在的大部分问题都只是从某一方面对均值不等式作探讨，而没有较为全面地详细论述均值不等式，在读者查阅文献时往往不能全面地获取所需的知识。

    均值不等式的起源悠久，应用广泛，在未来的学科发展应用中也有着广泛的前景，探索与研究均值不等式的意义不言而喻。

毕业论文关键词：  均值不等式 ; 几何思想 ; 数学归纳法 ; 排序原理 ;恒等变形     

Abstract Inequality theory is very momentous in the field of mathematics, and mean inequality is the core content of inequality theory, thus seriously studys on average inequality being considered very significant。 Application of the average inequality in elementary mathematics and advanced mathematics has been known as being important and extensive, so it needs a thorough understanding of mean inequality, and it is necessary for us to conduct a comprehensive study in-depth about mean inequality。 

The main contents of the research includes the form and nature of mean inequality, certification and popularization of mean inequality, applications of mean inequality,。and inquiry about transformation。 In this paper, the main research results in all aspects of mean inequality has been obtained。 

The main research results includes: the analysis of the source of mean inequality, methods of proof of mean inequality being pided into two categories(algebra and geometry), comparing the advantages and disadvantages of both at the same time, summarizing the generalization of average inequality in form and in nature。 Otherwise,   this paper has made a detailed discussion combined with examples about all kinds of mistakes when using the mean inequality。 

At present most of the problems of the study about mean inequality are inquiries of it's simple aspect , and there are hardly comprehensively discussions about the mean inequality。When readers literature ，they often can't obtain the required knowledge fully。 

Mean inequality has an origin of long history and wide application, also has a broad prospect on development and application of the disciplines in the future, the meaning of exploring and researching the mean inequality is self-evident。

Keywords：mean inequality ; geometrical thought ; mathematical induction ; collating principle ; identical deformation

目 录

第一章 均值不等式的基本概念 1

1。1均值不等式的来源与定义 1

1。1。1均值不等式的来源 1

1。1。2均值不等式的定义 1

1。2均值不等式的基本性质