# General form of Chebyshev type inequality for generalized Sugeno integral

## TL;DR

This paper establishes a comprehensive Chebyshev type inequality for generalized Sugeno integrals, extending previous results and including necessary and sufficient conditions involving m-positively dependent functions.

## Contribution

It introduces a general form of Chebyshev inequality for the generalized upper Sugeno integral, encompassing a broader class of functions and measures, and corrects prior inaccuracies.

## Key findings

- Derived necessary and sufficient conditions for the inequality
- Extended results to m-positively dependent functions
- Identified and corrected flaws in existing literature

## Abstract

We prove a~general form of Chebyshev type inequality for generalized upper Sugeno integral in the form of necessary and sufficient condition. A key role in our considerations is played by the~class of $m$-positively dependent functions which includes comonotone functions as a~proper subclass. As a~consequence, we state an equivalent condition for Chebyshev type inequality to be true for all comonotone functions and any monotone measure. Our results generalize many others obtained in the framework of q-integral, seminormed fuzzy integral and Sugeno integral on the real half-line. Some further consequences of these results are obtained, among others Chebyshev type inequality for any functions. We also point out some flaws in existing results and provide their improvements.