On the Minkowski-Hölder type inequalities for generalized Sugeno integrals with an application

Abstract

In this paper, we use a new method to obtain the necessary and sufficient condition guaranteeing the validity of the Minkowski-Hölder 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 [10] is not true. Next, we study the Minkowski-Hölder inequality for the lower Sugeno integral and the class of µsubadditive functions introduced in [18]. 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 in [5].

On $\star$-associated comonotone functions

“…For this reason, injective infimum preserving functions are characterized. Using a similar method, involving supremum and infimum preserving mappings, we also positively answer the second open problem stated by Boczek and Kaluszka in [1] about the existence of a pair of binary operations such that the generalized upper Sugeno integral equals the generalized lower Sugeno integral for each measurable function and each monotone measure. All the necessary notions and the corresponding result are included in Section 3.…”

“…During the way to provide a description of operations for which -associatedness implies comonotonicity we restate some basic observations from [1] from a more general (algebraic) point of view.…”

“…It can be easily seen that the operation min represents such a function, hence any two measurable functions are min-associated. Boczek and Kaluszka proved in [1] that the comonotonicity of real functions is equivalent to +-associatedness with + being the standard addition on the real line R. Then they formulated the following OPEN PROBLEM 1: Is there any non-trivial binary operation = + on Y such that the -associatedness is equivalent to the comonotonicity condition?…”

New monotone measure-based integrals inspired by scientific impact problem

“…We formulate properties which any integral should possess. 2) in [5]) under some additional restrictions.…”

“…Both functionals are integrals in the spirit of definition introduced here. Integrals have many applications, especially in multicriteria decision theory, economy, optimalization or data classification [5,10,12,14,22,23,38,39,44]. To the best of our knowledge, there are only very few papers which describe the connection between integrals and scientometric indices, see [21,39].…”