Abstract Being an important part of classical analysis, Jensen's inequality has drawn much attention recently. Due to its generality, the inequality based on non-additive integrals appears in many forms, such as Sugeno integrals, Choquet integrals and pseudo-integrals. As a well-known generalization of classical one, the set-valued analysis is frequently applied to the research of mathematical economy, control theory and so on. Thus, it is of great necessity to generalize the set-valued case. Motivated by the pioneering work of Costa's Jensen's fuzzy-interval-valued inequality and Strboja et al.'s Jensen's set-valued inequality based on Aumann integrals and pseudo-integrals respectively, this paper focuses particularly on proving certain kinds of Jensen's set-valued inequalities and fuzzy set-valued inequalities. These inequalities consist of two families: the related convex (or concave) function is a set-valued or fuzzy set-valued function and the integrand is a real-valued function; the related convex (or concave) function is a real-valued function and the integrand is a set-valued or fuzzy set-valued function. Particularly, Jensen's interval-valued and fuzzy-interval-valued inequalities, including Costa's, are obtained as corollaries.

中文翻译：

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集值函数和模糊集值函数的 Jensen 不等式

摘要 作为经典分析的重要组成部分，Jensen 不等式最近备受关注。由于其普遍性，基于非可加积分的不等式以多种形式出现，如 Sugeno 积分、Choquet 积分和伪积分。集值分析作为经典理论的一种众所周知的推广，经常被应用于数理经济学、控制理论等研究中。因此，非常有必要推广集合值的情况。受Costa的Jensen模糊区间值不等式和Strboja等人分别基于Aumann积分和伪积分的Jensen集值不等式的开创性工作的启发，本文重点研究了某些Jensen集值不等式的证明和模糊集值不等式。这些不等式包括两个系列：相关的凸（或凹）函数是一个集值或模糊集值函数，被积函数是一个实值函数；相关的凸（或凹）函数是实值函数，被积函数是集值或模糊集值函数。特别是，Jensen 的区间值不等式和模糊区间值不等式，包括科斯塔的，是作为推论获得的。