On the Minkowski-H\"{o}lder type inequalities for generalized Sugeno integrals with an application

In this paper, we use a new method to obtain the necessary and sufficient condition guaranteeing the validity of the Minkowski-H\"{o}lder type inequality for the generalized upper Sugeno integral in the case of functions belonging to a wider class than the comonotone functions. As a by-product, we show that the Minkowski type inequality for seminormed fuzzy integral presented by Daraby and Ghadimi in General Minkowski type and related inequalities for seminormed fuzzy integrals, Sahand Communications in Mathematical Analysis 1 (2014) 9--20 is not true. Next, we study the Minkowski-H\"{o}lder inequality for the lower Sugeno integral and the class of $\mu$-subadditive functions introduced in On Chebyshev type inequalities for generalized Sugeno integrals, Fuzzy Sets and Systems 244 (2014) 51--62. The results are applied to derive new metrics on the space of measurable functions in the setting of nonadditive measure theory. We also give a partial answer to the open problem 2.22 posed by Borzov\'a-Moln\'arov\'a and et al in The smallest semicopula-based universal integrals I: Properties and characterizations, Fuzzy Sets and Systems 271 (2015) 1--17.Comment: 19 page

The Choquet integral as Lebesgue integral and related inequalities

summary:The integral inequalities known for the Lebesgue integral are discussed in the framework of the Choquet integral. While the Jensen inequality was known to be valid for the Choquet integral without any additional constraints, this is not more true for the Cauchy, Minkowski, Hölder and other inequalities. For a fixed monotone measure, constraints on the involved functions sufficient to guarantee the validity of the discussed inequalities are given. Moreover, the comonotonicity of the considered functions is shown to be a sufficient constraint ensuring the validity of all discussed inequalities for the Choquet integral, independently of the underlying monotone measure