凸函数及其在证明不等式中的应用37885

本科毕业论文

题目凸函数及其在证明不等式中的应用系别数学与信息科学学院

专业数学与应用数学

指导教师吴开腾

评阅教师

班级2004级2班

姓名冀学本

学号064

2008 年５月２７日

目录

摘要 ........................................................................................................................... 错误!未定义书签。Abstract ................................................................................................................... 错误!未定义书签。1引言 ....................................................................................................................... 错误!未定义书签。

2 凸函数的等价定义...................................................................................... 错误!未定义书签。

凸函数三种定义的等价性的讨论...................................................................... 错误!未定义书签。

定义1⇔定义2 ................................................................................................... 错误!未定义书签。

定义1⇔定义3................................................................................................... 错误!未定义书签。判定定理与JESEN不等式 .................................................................................... 错误!未定义书签。3．性质 ................................................................................................................... 错误!未定义书签。4凸函数在不等式证明中的应用............................................................. 错误!未定义书签。

利用凸函数定义证明不等式 ............................................................................... 错误!未定义书签。

利用凸函数性质证明不等式 ............................................................................. 错误!未定义书签。结束语...................................................................................................................... 错误!未定义书签。参考文献 ................................................................................................................ 错误!未定义书签。致谢 ........................................................................................................................... 错误!未定义书签。

摘要首先给出了凸函数的三个典型定义，分析了它们之间的关系，并证明了三种定义之间的等价性．接着给出了凸函数的一个判定定理以及Jesen不等式．然后讨论了凸函数的几条常用性质，通过例题展示了凸函数在不等式证明中的应用．凸函数具有重要的理论研究价值和实际广泛应用，利用凸函数的性质证明不等式；很容易证明不等式的正确性．因此，正确理解凸函数的定义、性质及应用，更对有关学术问题进行推广研究起着举足轻重的作用．在不等式证明中的应用并举例说明解题思路与证明方法，最后证明了几个常见的重要不等式．并得到了几种常用凸函数的形式．

关键词凸函数，凸性不等式，jensen不等式

Abstract First has given the convex function three model definition，has analyzed between them the relations，and has proven between three kind of definition equivalence. Then has given a convex function determination theorem as well as the Jesen inequality. Then discussed convex function several commonly used nature，has demonstrated the convex function in inequality proof application through the sample question. The convex function has the important fundamental research value and the actual widespread application，the use convex function nature proof inequality;Very easy to prove the inequality the accuracy. Therefore，the correct understanding convex function's definition，the nature and the application，carry on the promotion to the related academic question to study the pivotal function. In the inequality proved that the application and explains with examples the problem solving mentality and the certificate method，finally has proven several common important inequalities. And obtained several kind of commonly used convex function forms.

Key words Convex function，convexity inequality，jensen inequality