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**On the Minkowski-Hölder type inequalities for generalized Sugeno integrals with an application.**
*(English)*
Zbl 1389.26063

The upper and lower Sugeno integrals are related to the Ky Fan metric and they coincide. Once replacing the multiplication operation min considered in the Sugeno integral by some more general kind of multiplication, in particular by a semicopula, one obtains a generalized upper Sugeno and a generalized lower Sugeno integral which, in general, differ. In this paper, the authors focus on a version of the Minkowski-Hölder inequality for the above mentioned integrals. Note that the considered inequality was known earlier (for the Sugeno integral) when the considered integrands were comonotone. The authors consider a wider class of functions and thus their generalizations concern not only the type of integral, but also the link between the considered functions being integrated. As a by-product, some failure in literature is shown. Moreover, the obtained results allow to the authors to derive some new metrics on the space of measurable functions in the setting of nonadditive measure theory. Finally, a partial answer to the open Problem 2.22 from [J. Borzová-Molnárová et al., Fuzzy Sets Syst. 271, 18–30 (2015; Zbl 1374.28026)] is given.

Reviewer: Radko Mesiar (Bratislava)