Soft Computing

, Volume 23, Issue 24, pp 12991–12999 | Cite as

Choquet integrals of r-convex functions

  • Hongxia WangEmail author


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.


Choquet 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.

Compliance with ethical standards

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.

Ethical approval

This article does not contain any studies with human participants or animals performed by the author.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Statistics and Data ScienceHenan University of Economics and LawZhengzhouChina
  2. 2.Analysis and Research Center on Education and Statistic Data of Henan ProvinceZhengzhouChina

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