## Abstract

Let A = (a_{n, k})_{n, k ≥ 0} be a non-negative matrix. Denote by L_{p, q} (A) the supremum of those L satisfying the following inequality:(underover(∑, n = 0, ∞) (underover(∑, k = 0, ∞) a_{n, k} x_{k})^{q})^{1 / q} ≥ L (underover(∑, k = 0, ∞) x_{k}^{p})^{1 / p} (X ∈ ℓ_{p}, X ≥ 0) . In this paper, we focus on the evaluation of L_{p, p} (A^{t}) for a lower triangular matrix A, where 0 < p < 1. A Borwein-type result is established. We also derive the corresponding result for the case L_{p, p} (A) with - ∞ < p < 0. In particular, we apply them to summability matrices, the weighted mean matrices, and Nörlund matrices. Our results not only generalize the work of Bennett, but also provide several analogues of those given in [Chang-Pao Chen, Dah-Chin Lour, Zong-Yin Ou, Extensions of Hardy inequality, J. Math. Anal. Appl. 273 (1) (2002) 160-171] and [P.D. Johnson Jr., R.N. Mohapatra, D. Ross, Bounds for the operator norms of some Nörlund matrices, Proc. Amer. Math. Soc. 124 (2) (1996), Corollary on p. 544]. Our results also improve Bennett's results for some cases.

Original language | English |
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Pages (from-to) | 1284-1294 |

Number of pages | 11 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 341 |

Issue number | 2 |

DOIs | |

State | Published - 15 May 2008 |

## Keywords

- Cesàro matrices
- Lower bounds
- Nörlund matrices
- Summability matrices
- Weighted mean matrices