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Choquet integral Jensen’s inequalities for set-valued and fuzzy set-valued functions

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Abstract

This article attempts to establish Choquet integral Jensen’s inequality for set-valued and fuzzy set-valued functions. As a basis, the existing real-valued and set-valued Choquet integrals for set-valued functions are generalized, such that the range of the integrand is extended from \(P_{0}(R^{+})\) to \(P_{0}(R)\), the upper and lower Choquet integrals are defined, and the fuzzy set-valued Choquet integral is introduced. Then Jensen’s inequalities for these Choquet integrals are proved. These include reverse Jensen’s inequality for nonnegative real-valued functions, real-valued Choquet integral Jensen’s inequalities for set-valued functions, and two families of set-valued and fuzzy set-valued Choquet integral Jensen’s inequalities. One is that the related convex function is set-valued or fuzzy set-valued, and the integrand is real-valued, the other is that the related convex function is real-valued, and the integrand is set-valued or fuzzy set-valued. The obtained results generalize earlier works (Costa in Fuzzy Sets Syst 327:31–47, 2017; Zhang et al. in Fuzzy Sets Syst 404:178–204, 2021).

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Acknowledgements

The authors would like to express heartfelt thanks to Prof. Pap, the editors-in-chief and the unknown reviewers for their generous help, and appreciation to Professor Costa and Dr. Štrboja et al. for their pioneering work. This study was supported by the National Natural Science Fund of China (11271063) and the Natural Science Fund of Jilin Province (20190201014JC).

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Authors and Affiliations

  1. College of Mathematics, Changchun Normal University, Changchun, 130032, P. R. China

    Deli Zhang

  2. College of Science, Changchun University, Changchun, 130022, P. R. China

    Caimei Guo

  3. Department of Mathematics and Physics, North China Electric Power University, Beijing, 100080, P. R. China

    Degang Chen

  4. College of Mathematics, Tianjin Normal University, Tianjin, 300387, P. R. China

    Guijun Wang

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  1. Deli Zhang
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  3. Degang Chen
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  4. Guijun Wang
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DZ wrote the original draft. CG, GW, and DC contributed to writing, reviewing, and editing.

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Correspondence to Deli Zhang.

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Zhang, D., Guo, C., Chen, D. et al. Choquet integral Jensen’s inequalities for set-valued and fuzzy set-valued functions. Soft Comput 25, 903–918 (2021). https://doi.org/10.1007/s00500-020-05568-2

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