The fuzzy integral for monotone functions

In this paper, we give some optimal upper bounds for the Sugeno’s integral of monotone functions. More precisely, we show that: If g : [0, ∞) → [0, ∞) is a continuous and strictly monotone function, then the fuzzy integral value p=⨍0agdμ, with respect to the Lebesgue measure μ, verifies the following sharp inequalities:(a)g(a-p)⩾pfor the increasing case, and(b)g(p)⩾pfor the decreasing case. Moreover, we show that under adequate conditions, these optimal inequalities provides a powerful tool for solving fuzzy integrals. Also, some examples and application are presented.