On Hardy q-inequalities

TL;DR

This paper establishes sharp q-analogue Hardy inequalities involving q-integrals and operators, extends results with Riemann-Liouville operators, and derives new discrete Hardy and Copson inequalities using these techniques.

Contribution

It introduces new sharp q-inequalities of Hardy type, extends them with Riemann-Liouville operators, and applies the methods to obtain novel discrete inequalities.

Findings

Proved sharp q-Hardy inequalities with explicit constants.

Extended inequalities to include Riemann-Liouville operators.

Derived new discrete Hardy and Copson inequalities.

Abstract

Some q-analysis variants of Hardy type inequalities of the form \int_0^b (x^{\alpha-1} \int_0^x t^{-\alpha} f(t) d_qt)^p d_qx \leq C \int_0^b f^p(t) d_qt with sharp constant C are proved and discussed. A similar result with the Riemann-Liouville operator involved is also proved. Finally, it is pointed out that by using these techniques we can also obtain some new discrete Hardy and Copson type inequalities in the classical case.