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).

本文试图为集值函数和模糊集值函数建立Choquet积分Jensen不等式。作为基础，对现有的用于设定值函数的实值和设定值Choquet积分进行了概括，从而将被积数的范围从\（P_ {0}（R ^ {+}）\）扩展到\ （P_ {0}（R）\）定义上下Choquet积分，并引入模糊集值Choquet积分。然后证明了这些Choquet积分的Jensen不等式。这些包括非负实值函数的逆Jensen不等式，集值函数的实值Choquet积分Jensen不等式以及两个族的集值和模糊集值Choquet积分Jensen不等式。一种是相关凸函数是集值或模糊集值，被积是实值，另一种是相关凸函数是实值，而被积函数是集值或模糊集值。获得的结果概括了早期的工作（Costa在Fuzzy Sets Syst 327：31–47，2017; Zhang等人在Fuzzy Sets Syst 404：178–204，2021）。