凹凸函數及其在不等式證明中的應用

凹凸函數及其在不等式證明中的應用

凹凸函數及其在不等式證明中的應用

摘 要：凹凸函數是研究證明不等式的有力工具。本文首先列出凸函數的4個等價定義，通過定義證明推導出了凸函數若干新性質、定理，得到凸函數常用的1些判別方法。最后將這些結論應用到不等式的證明中去，使1些復雜的不等式問題迎刃而解，且利用凸函數來證明比其他的方法簡潔、巧妙。文中證明的1些經典不等式和1些與實際生活、生產相關的不等式，同時為數學競賽和初等數學構造1些不等式問題提供了理論依據，同時對人們的生活有1定的指導意義及參考價值。
關鍵詞：凹凸函數；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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