Abstract
The first power weighted version of Hardy’s inequality can be rewritten as $$ \int _{0}^{\infty } \biggl( x^{\alpha -1} \int _{0}^{x} \frac{1}{t ^{\alpha }}f(t)\,dt \biggr) ^{p}\,dx\leq \biggl[ \frac{p}{p-\alpha -1} \biggr] ^{p} \int _{0}^{\infty }f^{p}(x)\,dx,\quad f\geq 0,p\geq 1, \alpha < p-1, $$ where the constant $C= [ \frac{p}{p-\alpha -1} ] ^{p}$ is sharp. This inequality holds in the reversed direction when $0\leq p<1$ . In this paper we prove and discuss some discrete analogues of Hardy-type inequalities in fractional h-discrete calculus. Moreover, we prove that the corresponding constants are sharp.