We obtain sharp two-sided inequalities between -norms of functions and , where is the Hardy operator, is its dual, and is a nonnegative measurable function on In an equivalent form, it gives sharp constants in the two-sided relationships between -norms of functions and , where is a nonnegative nonincreasing function on with In particular, it provides an alternative proof of a result obtained by Kruglyak and Setterqvist (Proc Am Math Soc 136:2005-2013, 2008) for and by Boza and Soria (J Funct Anal 260:1020-1028, 2011) for all , and gives a sharp version of this result for 1 < p < 2.