摘要：凸函数是一类非常重要的函数，它的概念最早见于Jensen[1905]著述中。它在纯粹数学和应用数学的众多领域中具有非常广泛的应用。现已成为数学规划、对策论、数理经济学、变分学和最优控制等学科的理论基础和有力工具。本文从凸函数出发引出几种不等式，利用凸函数和不等式的关系求解数学问题，或由凸函数和一个不等式来推出其它不等式。不等式的应用，让我们能够选择合适便捷的方法来解决有关的数学问题。

关键词：凸函数；Jensen不等式；赫尔德不等式；柯西不等式；应用

Abstract:The convex function is a very important function, its concept first appeared in the Jensen [1905]. It has very wide application in many areas including the pure mathematics and the Applied Mathematics. It has now become the theoretical foundation and the powerful tool of  many subjects including with programming, game theory, mathematical economics, calculus of variations and optimal control. In this paper, we will get some inequalities frome the convex function and solve some mathematical problems using the relation between the convex function and the related inequality, or launch other inequalities by the convex function and inequality. The application of inequality enable us to choose some suitable and convenient methods to solve the related mathematical problems.

Key word: Convex function；Jensen inequality； Holder inequality ；Cauchy  inequality；application