Choquet integrals of r-convex functions

Abstract

This study explores the upper bound and the lower bound of Choquet integrals for r-convex functions. Firstly, we show that the Hadamard inequality of this kind of integrals does not hold, but in the framework of distorted Lebesgue measure we can provide the similar Hadamard inequality of Choquet integral as the r-convex function is monotone. Secondly, the upper bound of Choquet integral for general r-convex function is estimated, respectively, in the case of distorted Lebesgue measure and in the non-additive measure. Finally, we present two Jensen’s inequalities of Choquet integrals for r-convex functions, which can be used to estimate the lower bound of this kind, when the non-additive measure is submodular. What’s more, we provide some examples in the case of the distorted Lebesgue measure to illustrate all the results.