Hardy-type inequalities in fractional h-discrete calculus

Abstract

Abstract The first power weighted version of Hardy’s inequality can be rewritten as ∫0∞(xα−1∫0x1tαf(t)dt)pdx≤[pp−α−1]p∫0∞fp(x)dx,f≥0,p≥1,α<p−1, $$ \int _{0}^{\infty } \biggl( x^{\alpha -1} \int _{0}^{x} \frac{1}{t ^{\alpha }}f(t)\,dt \biggr) ^{p}\,dx\leq \biggl[ \frac{p}{p-\alpha -1} \biggr] ^{p} \int _{0}^{\infty }f^{p}(x)\,dx,\quad f\geq 0,p\geq 1, \alpha < p-1, $$ where the constant C=[pp−α−1]p $C= [ \frac{p}{p-\alpha -1} ] ^{p}$ is sharp. This inequality holds in the reversed direction when 0≤p<1 $0\leq p<1$. 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.

Keywords

• Inequality
• Integral operator
• h-calculus
• h-integral
• Discrete Fractional Calculus