Lower bound of four-dimensional Hausdorff matrices

Abstract

Abstract Let H=(hnmjk) $\mathsf{H}=(h_{nmjk})$ be a non-negative four-dimensional matrix. Denote by Lp(H) $L_{p}(\mathsf{H})$ the supremum of those ℓ satisfying the following inequality: (∑n=0∞∑m=0∞(∑j=0∞∑k=0∞hnmjkxj,k)p)1/p≥ℓ(∑j=0∞∑k=0∞xj,kp)1/p, $$ { \Biggl( {\sum_{n = 0}^{\infty }{\sum _{m = 0}^{\infty } {{{ \Biggl( {\sum _{j = 0}^{\infty }{\sum_{k = 0}^{\infty }{{h_{nmjk}} {x_{j,k}}} } } \Biggr)}^{p}}} } } \Biggr)^{1/p}} \ge \ell { \Biggl( {\sum_{j = 0}^{\infty }{\sum _{k = 0}^{ \infty }{x_{j,k}^{p}} } } \Biggr)^{1/p}}, $$ where x=(xj,k)∈Lp $x=(x_{j,k}) \in \mathcal{L}_{p}$ with xj,k≥0 $x_{j,k}\ge 0$. In this paper a Hardy type formula is established for Lp(Hμ×λt) $L_{p}(\mathsf{H}_{ \mu \times \lambda }^{t})$, where 0<p≤1 $0< p\le 1$ and Hμ×λ $\mathsf{H}_{\mu \times \lambda }$ is a four-dimensional Hausdorff matrix. A similar result is also obtained for the case in which Hμ×λ $\mathsf{H}_{\mu \times \lambda }$ is replaced by Hμ×λt $\mathsf{H}_{\mu \times \lambda }^{t}$. As a consequence, we apply the results to some special four-dimensional Hausdorff matrices such as Cesàro, Euler, Hölder and Gamma matrices. Our results contain some generalizations of Copson’s discrete inequality.

Keywords

• Four-dimensional Hausdorff matrix