Jensen's inequalities for pseudo-integral

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