基于r-凸函数的Choquet积分不等式

Choquet Integral Inequalities Based on r-convex Functions

本文研究了r-凸函数的Choquet积分的Hadamard不等式和詹森不等式。首先,针对单调r-凸函数,研究了其Choquet积分的类似Hadamard型不等式;接着,分别在扭曲勒贝格测度和非可加测度下,估计了r-凸函数的Choquet积分的上界;最后,在非可加测度是凹的情形下,给出了两个r-凸函数的Choquet积分的詹森不等式,其可用来估计Choquet积分的下界。另外,在扭曲勒贝格测度下,对文中所有结果进行了例证。

In this paper we investigate the Hadamard inequality and Jensen’s inequality of Choquet integral for r-convex functions. Firstly, for monotone r-convex function we state the similar Hadamard inequality of the Choquet integral. Secondly, we estimate the upper bound of Choquet integral for general r-convex function, respectively, in the ditorted Lebesgue measure and in the non-additive measure. Finally, we present two Jensen’s inequalities of Choquet integral for r-convex functions, which can be used to estimate the lower bound of this kind, when the non-additive measure is concave. We provide some examples in the framework of the distorted Lebesgue measure to illustrate all the results.