Choquet integrals of r-convex functions
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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.
KeywordsChoquet integral r-convex function Inequality
This work is supported by Scientific Research Foundation for Doctors of Henan University of Economics and Law, and by Philosophy and Social Science Planning Project of Henan Province, China.
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Conflict of interest
The author confirms that this work is original and has not been published elsewhere nor is it currently under consideration for publication elsewhere and there are no known conflicts of interest associated with this publication.
This article does not contain any studies with human participants or animals performed by the author.
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