Lower bounds of Copson type for Hausdorff matrices

Let 1 ⩽ p ⩽ ∞, 0 < q ⩽ p, and A = (an,k)n,k⩾0 ⩾ 0. Denote by Lp,q(A) the supremum of those L satisfying the following inequality:∑n=0∞∑k=0∞an,kxkq1/q⩾L∑k=0∞xkp1/p,whenever X={xn}n=0∞∈ℓp and X ⩾ 0. The purpose of this paper is to find the exact value of Lp,q(A) when A is a Hausdorff matrix or its transpose. In particular, we apply it to Cesàro matrices, Hölder matrices, Gamma matrices, and generalized Euler matrices. Our results generalize the work of Bennett.