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

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 P0(R+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{0}(R^{+})$$\end{document} to P0(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{0}(R)$$\end{document}, 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).