A strengthening of Jensen's inequality for asymmetric Choquet integral

Recently, Zhang et al. [Fuzzy Sets and Systems 430 (2022) 79-87, Fuzzy Sets and Systems 457 (2023) 119-124] proposed some Jensen's inequalities for asymmetric Choquet integral. Then Wang [Fuzzy Sets and Systems 457 (2023) 105-118] investigated some refinements of Jensen's inequalities for Choquet integral. In this paper, sharp inequalities related to Jensen for asymmetric Choquet integral are studied. Our results have many benefits such as applicability, simplicity, easy to use and require only knowledge of some important properties of asymmetric Choquet integral.