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凹凸函數(shù)及其在不等式證明中的應(yīng)用
凹凸函數(shù)及其在不等式證明中的應(yīng)用
摘 要：凹凸函數(shù)是研究證明不等式的有力工具。本文首先列出凸函數(shù)的4個(gè)等價(jià)定義，通過定義證明推導(dǎo)出了凸函數(shù)若干新性質(zhì)、定理，得到凸函數(shù)常用的1些判別方法。最后將這些結(jié)論應(yīng)用到不等式的證明中去，使1些復(fù)雜的不等式問題迎刃而解，且利用凸函數(shù)來證明比其他的方法簡(jiǎn)潔、巧妙。文中證明的1些經(jīng)典不等式和1些與實(shí)際生活、生產(chǎn)相關(guān)的不等式，同時(shí)為數(shù)學(xué)競(jìng)賽和初等數(shù)學(xué)構(gòu)造1些不等式問題提供了理論依據(jù)，同時(shí)對(duì)人們的生活有1定的指導(dǎo)意義及參考價(jià)值。
關(guān)鍵詞：凹凸函數(shù)；不等式證明；琴生(Jensen)不等式；赫爾德(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
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