Some q-inequalities for Hausdorff operators

Abstract

We calculate the sharp bounds for some q-analysis variants of Hausdorff type inequalities of the form

{\displaystyle \int \begin{array}{l}+\infty \\ \mathrm{0}\end{array}}{\left({\displaystyle \int \begin{array}{l}+\infty \\ \mathrm{0}\end{array}}{\displaystyle \frac{\varphi (t)}{t}}f{\displaystyle \frac{x}{t}}{{\displaystyle d}}_{q}t\right)}^p{{\displaystyle d}}_{q}x\le {{\displaystyle C}}_{\varphi }{\displaystyle \int \begin{array}{l}b\\ \mathrm{0}\end{array}}{f}^p(t){{\displaystyle d}}_{q}t.

As applications, we obtain several sharp q-analysis inequalities of the classical positive integral operators, including the Hardy operator and its adjoint operator, the Hilbert operator, and the Hardy-Littlewood-Pólya operator.

Key words： Sharp constant; Hausdorff operator; Hilbert operator; q-inequality

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