Lower bounds of Copson type for Hausdorff matrices (Q865462)

Let \(A=\left( a_{m,n}\right) _{m,n\geq 0}\) be a nonnegative matrix. For \( 1\leq p\leq \infty \) and \(0<q\leq p\) let \(L_{p,q}(A)\) be the supremum of the numbers \(L\) satisfying the inequality: \[ \left( \sum_{m=0}^{\infty }\left( \sum_{n=0}^{\infty }a_{m,n}x_{n}\right) ^{q}\right) ^{1/q}\geq L\left( \sum_{n=0}^{\infty }x_{n}^{p}\right) ^{1/p}, \] whenever \(\left\{ x_{n}\right\} _{n=0}^{\infty }\in \ell ^{p}\) with \( x_{n}\geq 0\) for all \(n\geq 0\). The authors give the exact value of \( L_{p,q}(A)\) in the case when \(A\) is a Hausdorff matrix. Applications to Cesàro matrices, Hölder matrics, gamma matrices, and generalized Euler matrices are given.