Improved Discrete Dual $p$-Hardy and Weighted Discrete $p$- Birman Inequalities

In this paper, we establish a new version of one dimensional generalized discrete dual $p$-Hardy inequality with a shift. Using this generalized discrete dual p-Hardy inequality, we obtain improvements of two discrete dual $p$-Hardy inequalities. To be specific, for $p>1$ and $A\in C_c({\mathbb{N}}_0)$ satisfying $A_0=A_1=0$, we first improve the discrete dual p-Hardy inequality \begin{align*} &\displaystyle\sum_{n=2}^{\infty}(n-1)^{p}| A_{n}-A_{n-1}|^{p}\geq\frac{1}{p^{p}}\displaystyle\sum_{n=2}^{\infty}|A_{n}|^{p}, \end{align*} where the associate constant term is sharp. Subsequently, we improve its power-type weighted discrete dual p-Hardy extension \begin{align*} &\displaystyle\sum_{n=2}^{\infty}(n-1)^α|A_{n}-A_{n-1}|^{p}\geq\Big(\frac{α+1-p}{p}\Big)^{p} \displaystyle\sum_{n=2}^{\infty}\frac{|A_{n}|^{p}}{n^{p-α}} \end{align*} for $p-1<\alpha \le p$, where the associated constant term is also sharp. We also establish a discrete $p$- Birman inequality with power weights. Furthermore, we establish a multivariable dual $p$-Hardy inequality with a sharp constant. The proof proceeds by first establishing the inequality for two variables and then extending the argument to multiple variables, while preserving the sharpness of the constant.