Suppose that \(\{A_{n}\}_{n=1}^{\infty }\) is a sequence of complex numbers, which is finitely supported on \(\mathbb {N}_{0}\) with \(A_{0}=0\) . We prove that the power-type weighted discrete Hardy’s inequality \(\begin{aligned}&\displaystyle \sum _{n=1}^{\infty }n^{\alpha }|A_{n}-A_{n-1}|^{3}\ge \Big (\frac{2-\alpha }{3}\Big )^{3}\displaystyle \sum _{n=1}^{\infty }\frac{|A_{n}|^{3}}{n^{3-\alpha }}, ~~\alpha \in [0, 2) \end{aligned}\) with the sharp constant \(\big (\frac{2-\alpha }{3}\big )^{3}\) is not optimal. It is demonstrated that an improvement of the aforesaid inequality is possible for \(\alpha \in [0, 1]\) . More precisely, we establish the following inequality \(\begin{aligned}&\displaystyle \sum _{n=1}^{\infty }n^{\alpha }|A_{n}-A_{n-1}|^{3}\ge \displaystyle \sum _{n=1}^{\infty }\rho _n(\alpha , \beta )|A_{n}|^{3}>\Big (\frac{2-\alpha }{3}\Big )^{3}\displaystyle \sum _{n=1}^{\infty }\frac{|A_{n}|^{3}}{n^{3-\alpha }}, ~\beta =\frac{2-\alpha }{3}, \end{aligned}\) where \(\rho _n(\alpha , \beta )\) is an improved weight such that \(\rho _n(\alpha , \beta )>\Big (\frac{2-\alpha }{3}\Big )^{3}n^{\alpha -3}\) holds point-wise for each \(n\in \mathbb {N}\) and \(\alpha \in [0, 1]\) .
