Optimal discrete $p$-Hardy-Rellich-Birman inequalities

We present a theory for constructing optimal lower bounds for the discrete half-line $p$-Laplacian of higher order $\ell\in\mathbb{N}$ and general $p>1$. The abstract framework introduces higher-order monotonicity and asymptotic constraints on a parameter sequence that determines optimal weights. As a concrete application, we specialize the parameter sequence to deduce new optimal discrete $p$-Hardy ($\ell=1$), $p$-Rellich ($\ell=2$), and $p$-Birman ($\ell\geq 3$) inequalities.