Jensen's inequalities for pseudo-integrals

In this paper, we introduce a general (⊕, ⊗)-convex function based on semirings ([a, b], ⊕, ⊗) with pseudo-addition ⊕ and pseudo-multiplication ⊗. The generalization of the finite Jensen’s inequality, as well as pseudo-integral with respect to (⊕, ⊗)-convex functions, is obtained. This also generalizes Jensen’s inequalities for Lebesgue integral and the results of Pap and Strboja [12]. Meanwhile, we also prove Jensen’s inequalities for pseudo-integrals on semirings ([ ˇ a, b],sup, ⊗) with respect to nondecreasing functions and present corresponding results for generalized fuzzy integrals.