TY - JOUR

T1 - Lower bounds of Copson type for the transposes of lower triangular matrices

AU - Chen, Chang Pao

AU - Wang, Kuo-Zhong

PY - 2008/5/15

Y1 - 2008/5/15

N2 - Let A = (an, k)n, k ≥ 0 be a non-negative matrix. Denote by Lp, q (A) the supremum of those L satisfying the following inequality:(underover(∑, n = 0, ∞) (underover(∑, k = 0, ∞) an, k xk)q)1 / q ≥ L (underover(∑, k = 0, ∞) xkp)1 / p (X ∈ ℓp, X ≥ 0) . In this paper, we focus on the evaluation of Lp, p (At) 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 Lp, 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.

AB - Let A = (an, k)n, k ≥ 0 be a non-negative matrix. Denote by Lp, q (A) the supremum of those L satisfying the following inequality:(underover(∑, n = 0, ∞) (underover(∑, k = 0, ∞) an, k xk)q)1 / q ≥ L (underover(∑, k = 0, ∞) xkp)1 / p (X ∈ ℓp, X ≥ 0) . In this paper, we focus on the evaluation of Lp, p (At) 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 Lp, 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.

KW - Cesàro matrices

KW - Lower bounds

KW - Nörlund matrices

KW - Summability matrices

KW - Weighted mean matrices

UR - http://www.scopus.com/inward/record.url?scp=39149084100&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2007.11.025

DO - 10.1016/j.jmaa.2007.11.025

M3 - Article

AN - SCOPUS:39149084100

VL - 341

SP - 1284

EP - 1294

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -