We establish an improvement of the power-type weighted discrete variant of Hardy’s inequality in one dimension. To be specific, we establish the following inequality \(\begin{aligned} \displaystyle \sum _{n=2}^{\infty }(n-1)^{\alpha }|A_{n}-A_{n-1}|^{2}&\ge \displaystyle \sum _{n=2}^{\infty }\beta _{n}(\alpha )|A_{n}|^{2}> \frac{(\alpha -1)^{2}}{4}\displaystyle \sum _{n=2}^{\infty }\frac{|A_{n}|^{2}}{n^{2-\alpha }}, ~~A_0=A_1=0, \end{aligned}\) where \(\alpha \in (1, 2]\) and \(\beta _{n}(\alpha )\) for \(2\le n\in {\mathbb {N}}\) is an improved weight. In addition, it is shown that the improved weight sequence \(\beta _{n}(\alpha )\) is critical. Further, we study some fundamental structures of the sequence space \(\Sigma _p\) for \(1<p<\infty \) originated from the improved inequality.
