Sharp bounds of Jensen type for the generalized Sugeno integral

TL;DR

This paper establishes sharp two-sided bounds of Jensen type for the generalized Sugeno integral applicable to all measurable functions, extending previous results and providing new inequalities in the context of real-valued functions.

Contribution

It introduces the first comprehensive bounds of Jensen type for the generalized Sugeno integral for any measurable function, extending prior specific cases.

Findings

Extended Jensen bounds to all measurable functions.

Corrected previous results of Abbaszadeh et al.

Derived sharp inequalities for symmetric Grabisch integral.

Abstract

In this paper we provide two-sided attainable bounds of Jensen type for the generalized Sugeno integral of {\it any} measurable function. The results extend the previous results of Rom\'an-Flores et al. for increasing functions and Abbaszadeh et al. for convex and concave functions. We also give corrections of some results of Abbaszadeh et al. As a by-product, we obtain sharp inequalities for symmetric integral of Grabisch. To the best of our knowledge, the results in the real-valued functions context are presented for the first time here.