The authors study the weighted
-boundedness of the multi-dimensional Hardy type operators in the generalized Morrey spaces
defined by an almost increasing function
and radial type weight
from the Bary-Stechkin-type class. Sufficient conditions are obtained, in terms of some integral inequalities imposed on
, for such a
-boundedness. In the case of local spaces the obtained conditions are also necessary. These results are applied to derive a similar weighted
-boundedness of the Riesz potential operator.