We consider and a corresponding geometric mean operator.E.T.
Sawyer showed that the Hardy-type goat guns nz inequality could be characterized by three independent conditions on the weights.We give a simple proof of the fact that if the weight is of product type, then in fact only one condition is needed.Moreover, by using this information and by performing a limiting procedure we can derive a weight characterization of the corresponding two-dimensional arteriors raphael chandelier Pólya-Knopp inequality with the geometric mean operator involved.