Bounds for the operator norms of some Nörlund matrices

Suppose $(p_n)_{n \geq 0}$ is a non-increasing sequence of non-negative numbers with $p_0 = 1$, $P_n = \sum _{j=0}^n p_j$, $n = 0, 1 \dots$, and $A = A(p_n) = (a_{nk})$ is the lower triangular matrix defined by $a_{nk} = p_{n-k} / P_n$, $0 \leq k \leq n$, and $a_{nk} = 0$, $n < k$. We show that the operator norm of $A$ as a linear operator on $\ell _p$ is no greater than $p / (p-1)$, for $1 < p < \infty$; this generalizes, yet again, Hardy's inequality for sequences, and simplifies and improves, in this special case, more generally applicable results of D. Borwein, Cass, and Kratz. When the $p_n$ tend to a positive limit, the operator norm of $A$ on $\ell _p$ is exactly $p/(p-1)$. We also give some cases when the operator norm of $A$ on $\ell _p$ is less than $p/(p-1)$.