Hardy-type Inequalities via Auxiliary Sequences

k=1 |ak|, in which C = (cj,k) and the parameter p are assumed fixed (p > 1), and the estimate is to hold for all complex sequences a. The lp operator norm of C is then defined as the p-th root of the smallest value of the constant U : ||C||p,p = U 1 p . Hardy’s inequality thus asserts that the Cesáro matrix operator C, given by cj,k = 1/j, k ≤ j and 0 otherwise, is bounded on lp and has norm ≤ p/(p− 1). (The norm is in fact p/(p − 1).) We say a matrix A is a summability matrix if its entries satisfy: aj,k ≥ 0, aj,k = 0 for k > j and ∑j k=1 aj,k = 1. We say a summability matrix A is a weighted mean matrix if its entries satisfy: aj,k = λk/Λj , 1 ≤ k ≤ j; Λj = j