一种二级凸函数

A CONVEX FUNCTION OF ORDER 2

本文研究林景荣定义的一种二级函数的一些理论,如充要条件、极值分布等等。同时,初探这种函数与Fan-Levison不等式、全正函数的联系。

A bounded function f(x) is said to be convex of order 2 on an inteival I if, for all XL x2 and x3, a≤x, <x2<x3≤b, we haveIf strict inequality holds in (1), then we say that f(x) is strictly convex of order 2.In this paper, first we investigate some basic properties of convex functions of order 2, e.g. a necessary and sufficient conditio that f(x) is convex of order 2 is thak f(x) can be expressed in the form of the indefinite integral of an absolutely continuous convex function,the distribution of extremums for strictly convex function of order 2 is also considered. Then, in Section 3, these functions are discussed in connection with the theory of total positive functions and Fan-Levinson inequality.