摘要：凸函数是一类比较重要的函数，它在纯粹数学和应用数学的众多领域中具有广泛应用，现已成为数学规划、控制论、函数论、泛函分析、最优化理论等领域的理论基础和有力工具。本文对凸函数的定义和相互关系、运算性质及分析性质做出较详尽的归纳总结，给出几个凸函数的判别定理，并利用这些特殊性质和定理证明两个重要不等式：Jensen不等式和Young不等式，讨论了一些初等不等式、函数不等式和积分不等式的应用。

关键词：凸函数；Jensen不等式；Young不等式；积分不等式

Abstract：Convex function is a kind of important function, it has broad applications in many areas of pure and applied mathematics. Now, convex function has become theoretical foundation and a powerful tool of mathematical programming, control theory, function theory, functional analysis, optimization theory and other fields. In this paper , I make more detailed summary about the definition and mutual relationship, operation properties and analysis of properties of convex function, and give several judgment theorems of convex function. I also take advantage of these special properties and theorems to prove two important inequalities: Jensen inequality and Young inequality, and discuss some elementary inequalities, function inequalities and integral inequalities application.

Key words: convex function; Jensen inequality; Young inequality; integral inequality