Hölder-Minkowski type inequality for generalized Sugeno integral

Abstract We give a necessary and sufficient condition for validity of the Holder-Minkowski type inequality for the generalized Sugeno integral, any monotone measure and the class of ⋆-associated functions. We also show a connection between integral inequalities and the problem of finding binary operations satisfying the distributivity inequalities. We characterize all semicopulas satisfying Holder and Minkowski inequalities for the seminormed fuzzy integral. One result has been obtained for the recently introduced q-integral. We describe the relationships among the existing sufficient conditions and show that our result generalizes most known Holder-Minkowski type inequalities for comonotone functions in the available literature. We also provide two other methods to get the above-mentioned inequality for the Sugeno integral and the class of m-subadditive functions as well as for the Sugeno integral of [ 0 , 1 ] -valued functions using a duality approach. We point out some flaws and correct the corresponding false statements appearing in the literature. At the end, we pose several problems and questions for further research.