凸函数与不等式

Function of convexity and inequality

凸性是一种重要的几何性质,凸函数是一种性质特殊的函数,凸集和凸函数在泛函分析、最优化理论、数理经济学等领域都有着广泛的应用.凸函数也是高等数学中的一个基本内容,它在证明比较复杂的不等式方面有着重大作用.探讨了凸函数与不等式之间的密切关系,利用函数的凸性来研究不等式,比传统方法更简洁,还进一步探讨了Jensen不等式的一些具体应用.

Convexity is an important geometric property, The convex function is a kind of function with special properties. Convexity and convex function have extensive applications in functional analysis, optimal theory, and mathematical economy. Convex function is also a basic content of higher maths, It plays an important role in proving more complex inequality. In this paper, a relation of convex function and inequality of Jensen is studied, it provides an easy way by using the convexity of function to study the inequality and discuss some typical case of inequality of Jensen.