Discrete weighted Hardy Inequality in 1-D

In this paper we consider a weighted version of one dimensional discrete Hardy's Inequality on half-line with power weights of the form $n^\alpha$. Namely we consider: \begin{equation} \sum_{n=1}^\infty |u(n)-u(n-1)|^2 n^\alpha \geq c(\alpha) \sum_{n=1}^\infty \frac{|u(n)|^2}{n^2}n^\alpha \end{equation} We prove the above inequality when $\alpha \in [0,1) \cup [5,\infty)$ with the sharp constant $c(\alpha)$. Furthermore when $\alpha \in [1/3,1) \cup \{0\}$ we prove an improved version of the above inequality. More precisely we prove \begin{equation} \sum_{n=1}^\infty |u(n)-u(n-1)|^2 n^\alpha \geq c(\alpha) \sum_{n=1}^\infty \frac{|u(n)|^2}{n^2} n^\alpha + \sum_{k=3}^\infty b_k(\alpha) \sum_{n=2}^\infty \frac{|u(n)|^2}{n^k}n^\alpha. \end{equation} for non-negative constants $b_k(\alpha)$.