凸函数的几种不同定义及作用

SEVERAL DIFFERENT DEFINITIONS AND FUNCTIONS OF CONVEX FUNCTION

凸函数是一类比较重要的函数,在数学规划中有着广泛的应用,考虑到凸函数与连续性、可导性之间的联系及凸函数在不等式证明方面的作用和意义,本文提出了凸函数的几种不同定义,并讨论了它们之间的等价性及凸函数的有关性质和它在不等式方面的相关应用。由于上凸函数和下凸函数统称为凸函数,所以本文所讨论的凸函数都是指下凸函数。

Convex function is an important kind of function and it is widely used in mathematic programming. Considering the relationship beteen continuity and derivative of convex function and their importance of application in proof of some inequalities,several kind of equal definitions of convex function are given in this paper, Their equality,some relative nature and their application in equalities are also discussed . As convex function from above and convex function from below are also called convex function,convex function discussed in this paper refers to convex function from below.