q-Hardy type inequalities for quantum integrals

Abstract

Abstract The aim of this work is to obtain quantum estimates for q-Hardy type integral inequalities on quantum calculus. For this, we establish new identities including quantum derivatives and quantum numbers. After that, we prove a generalized q-Minkowski integral inequality. Finally, with the help of the obtained equalities and the generalized q-Minkowski integral inequality, we obtain the results we want. The outcomes presented in this paper are q-extensions and q-generalizations of the comparable results in the literature on inequalities. Additionally, by taking the limit q → 1 − $q\rightarrow 1^{-}$ , our results give classical results on the Hardy inequality.

Keywords

• Hardy inequality
• Opial inequality
• Hölder’s inequality