Some new refined multidimensional Hardy-type inequalities for $p\geq 2$ and their duals are derived and discussed. Moreover, these inequalities hold in the reversed direction when $p \neq 1, 1\leq p\leq 2.\ $The results obtained are based mainly on some new results for superquadratic and subquadratic functions. In particular, our results further extend the recent results in [J.A. Oguntuase and L.-E. Persson, Refinement of Hardy's inequalities via superquadratic and subquadratic functions, J. Math. Anal. Appl., 339 (2008), no. 2, 1305-1312] to a multidimensional setting.
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