Geodesic
Some characterizing conditions for the Hardy inequality
Eurasian mathematical journal, Tome 1 (2010) no. 2, pp. 86-98

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In this paper, a new scale of necessary and sufficient conditions for the validity of the Hardy and reverse Hardy inequalities in the cases $$ 0\frac{q}{p}1,\qquad p\in(-\infty,0)\cup(1,\infty), $$ and $$ 0\frac{p}{q}1,\qquad q\in(0,1) $$ is found and estimates for the best constants are derived. 
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A. Kufner; K. Kuliev. Some characterizing conditions for the Hardy inequality. Eurasian mathematical journal, Tome 1 (2010) no. 2, pp. 86-98. http://geodesic.mathdoc.fr/item/EMJ_2010_1_2_a5/