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

TL;DR

This paper establishes necessary and sufficient conditions for Minkowski-Hölder inequalities for generalized Sugeno integrals, extends the class of functions considered, and applies results to define new metrics in nonadditive measure spaces.

Contribution

It provides a new method for Minkowski-Hölder inequalities for generalized Sugeno integrals beyond comonotone functions and addresses open problems in the field.

Findings

Necessary and sufficient conditions for Minkowski-Hölder inequalities established.

The Minkowski inequality for seminormed fuzzy integrals is shown to be false.

New metrics on measurable functions space are derived using these inequalities.

Abstract

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.