Steffensen type inequalities for fuzzy integrals

We provide several Steffensen type inequalities for the Sugeno integral. The inequalities are of the form −∫abfgdμ⩽A−∫aa+λfdμ+Bor−∫abfgdμ⩽A−∫b−λbfdμ+B,where A, B are constants, μ   is a fuzzy measure on R,R,g: [a, b  ] → [0, 1], f:[a,b]→R+f:[a,b]→R+ is nonincreasing or nondecreasing and λ=(b−a)∧−∫abgdμ. We show that some of our sufficient conditions are also necessary.