On a functional inequality stemming from the Gr\"uss-type inequalities

TL;DR

This paper investigates a new functional inequality related to generalized Grüss inequalities, exploring bounds involving a complex-valued functional and a real-valued map, with potential implications for inequality theory.

Contribution

It introduces a novel inequality involving a complex functional and a real map, extending the scope of Grüss-type inequalities.

Findings

Derived bounds for the functional inequality

Generalized Grüss inequality applications

Potential for broader inequality frameworks

Abstract

We study functional inequality of the form $$|T(f,h)-T(f,g)T(g,h)| \leq F(f,g)F(g,h) -F(f,h)$$ where $T$ is a complex-valued functional and $F$ is a real-valued map. Motivation for our studies comes from some generalizations of Gr\"uss inequality.