On Cauchy–Schwarz’s inequality for Choquet-like integrals without the comonotonicity condition

Abstract

Cauchy-Schwarz’s inequality is one of the most important inequalities in probability, measure theory and analysis. The problem of finding a sharp inequality of Cauchy–Schwarz type for Sugeno integral without the comonotonicity condition based on the multiplication operator has led to a challenging and an interesting subject for researchers. In this paper, we give a Cauchy–Schwarz’s inequality without the comonotonicity condition based on pseudo-analysis for two classes of Choquet-like integrals as generalizations of Choquet integral and Sugeno integral. In the first class, pseudo-operations are defined by a continuous strictly increasing function $g$. Another class concerns the Choquet-like integrals based on the operator “$\sup$” and a pseudo-multiplication $\otimes$. When working on the second class of Choquet-like integrals, our results give a new version of Cauchy–Schwarz’s inequality for Sugeno integral without the comonotonicity condition based on the multiplication operator.