凹凸函数及其在不等式证明中的应用

凹凸函数及其在不等式证明中的应用

摘 要：凹凸函数是研究证明不等式的有力工具。本文首先列出凸函数的4个等价定义，通过定义证明推导出了凸函数若干新性质、定理，得到凸函数常用的1些判别方法。最后将这些结论应用到不等式的证明中去，使1些复杂的不等式问题迎刃而解，且利用凸函数来证明比其他的方法简洁、巧妙。文中证明的1些经典不等式和1些与实际生活、生产相关的不等式，同时为数学竞赛和初等数学构造1些不等式问题提供了理论依据，同时对人们的生活有1定的指导意义及参考价值。

关键词：凹凸函数；不等式证明；琴生(Jensen)不等式；赫尔德(Holder)不等式；柯西(Cauchy)不等式。

Concave-convex function and its application in proving inequalities

Abstract: Concave-convex function is a powerful tool to study and prove the inequality. This article firstly lists four equivalent definitions of the convex function and deduces some new properties and theorems of the convex function through proving the definitions, so as to obtain some distinctive methods of the convex function which are used very frequently. Finally these conclusions will be applied to prove the inequality, so that they can make some complex inequality questions to be easily solved. And also using the convex function to prove inequality is more terse and ingenious than others. Some classical inequalities and some real life, production-related inequalities which are proved in this article provide the theory basis of structure some inequality questions to the mathematics competition and the elementary mathematics. Meanwhile they have the instruction significance and reference value to people’s life.

Key words: Concave-convex function; Inequality proof; Jensen inequality; Holder inequality; Cauchy inequality