Extensions of the Inequalities of Hardy and Hilbert

TL;DR

This paper generalizes the classical Hardy and Hilbert inequalities, demonstrating their equivalence through duality methods and providing examples to show the non-triviality of these extensions.

Contribution

It introduces generalized versions of Hardy and Hilbert inequalities and proves their equivalence using H^1-BMOA duality.

Findings

Generalized Hardy and Hilbert inequalities established

Proved the equivalence of the generalized inequalities

Provided examples demonstrating the non-triviality of the extensions

Abstract

In this note we produce generalized versions of the classical inequalities of Hardy and of Hilbert and we establish their equivalence. Our methods rely on the H^1-BMOA duality. We produce a class of examples to establish that the generalizations are non-trivial.