Some q-inequalities for Hausdorff operators

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

$$int_0^{ + infty } {{{left( {int_0^{ + infty } {frac{{phi left( t right)}}{t}fleft( {frac{x}{t}} right){d_q}t} } right)}^p}{d_q}x} leqslant {C_phi }int_0^b {{f^p}left( t right)} {d_q}t$$

. 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.