A new refined weighted Hardy inequality for p 2 is proved and discussed. The inequality is reversed for 1 < p 2, which means that for p = 2 we have equality. The main tool in the proofs are some new results for superquadratic and subquadratic functions.

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Some new multidimensional Hardy-type inequalities involving arithmetic mean operators with general positive kernels are derived. Our approach is mainly to use a convexity argument and the results obtained improve some known results in the literature and, in particular, some recent results in [S. Kaijser, L. Nikolova, L.-E. Persson, A. Wedestig, Hardytype inequalities via convexity, Math. Inequal. Appl. 8 (3) (2005) 403-417] are generalized and complemented.