Sharp Hermite-Hadamard, Fejér and Discrete Jensen-Type Inequalities for Fuzzy-Number-Valued h-Preinvex Mappings via the Choquet Integral
DOI:
https://doi.org/10.31185/bsj.Vol23.Iss46.1841Keywords:
Choquet integral; fuzzy number; h-preinvex mapping; Hermite-Hadamard inequality; Fejér inequality; capacity; non-additive measureAbstract
Background: Hermite-Hadamard inequalities are central estimates for convex and generalized convex mappings. Their extension to fuzzy-number-valued mappings under a non-additive capacity is non-trivial because the Choquet integral is monotone and positively homogeneous but not additive in general.
Materials and Methods: This paper develops a level-wise approach based on fuzzy numbers, the Kulisch-Miranker order on all level sets, invex paths satisfying Condition C, and normalized symmetric submodular capacities. A level-wise Choquet integral is used so that every fuzzy inequality is reduced to two scalar Choquet inequalities at each level.
Results: New capacity-stable Hermite-Hadamard bounds, a weighted Hermite-Hadamard-Fejér estimate, and a discrete Choquet-Jensen-type upper estimate are established for nonnegative fuzzy-number-valued h-preinvex mappings. The constants are expressed through Choquet integrals of and of the weight function; in the additive case they reduce to the classical integral constants. Equality is obtained for affine fuzzy-number-valued mappings when and the capacity is Lebesgue additive.
Conclusion: The results unify fuzzy-number-valued convexity, preinvexity, and non-additive integration in a framework that avoids the invalid additivity assumptions often associated with Choquet-type inequalities.
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