Optimal relationships between $$L^p$$ -norms for the Hardy operator and its dual

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We obtain sharp two-sided inequalities between $$L^p$$ -norms $$(1<p<\infty )$$ of functions $$\textit{Hf}$$ and $$H^*f$$ , where $$H$$ is the Hardy operator, $$H^*$$ is its dual, and $$f$$ is a nonnegative measurable function on $$(0,\infty ).$$ In an equivalent form, it gives sharp constants in the two-sided relationships between $$L^p$$ -norms of functions $$H\varphi -\varphi $$ and $$\varphi $$ , where $$\varphi $$ is a nonnegative nonincreasing function on $$(0,+\infty )$$ with $$\varphi (+\infty )=0.$$ In particular, it provides an alternative proof of a result obtained by Kruglyak and Setterqvist (Proc Am Math Soc 136:2005–2013, 2008) for $$p=2k \,\,(k\in \mathbb N )$$ and by Boza and Soria (J Funct Anal 260:1020–1028, 2011) for all $$p\ge 2$$ , and gives a sharp version of this result for $$1<p<2$$ .