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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:341

Article

Chen, Chang-Pao¹  Wang, Kuo-Zhong¹ 

[1] Hsuan Chuang Univ, Dept Appl Math, Hsinchu 300, Taiwan

关键词: lower bounds;    summability matrices;    weighted mean matrices;    Norlund matrices;    Cesaro matrices;   

DOI  :  10.1016/j.jmaa.2007.11.025

来源: Elsevier

【 摘 要 】

Let A= (a(n,k))(n,k)>= 0 be a non-negative matrix. Denote by L-p,L-q(A) the supremum of those L satisfying the following inequality: (Sigma(infinity)(n=0)(Sigma(infinity)(k=0)a(n,k)x(k))(q))(1/q) >= L(Sigma(infinity)(k=0)x(k)(p))(1/p) (X epsilon l(p), X >= 0). In this paper, we focus on the evaluation of L-p,L-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,L-p(A) with -infinity < p < 0. In particular, we apply them to summability matrices, the weighted mean matrices, and Norlund 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 On, 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 Norlund matrices, Proc. Amer. Math. Soc. 124 (2) (1996), Corollary on p. 544]. Our results also improve Bennett's results for some cases. (C) 2007 Elsevier Inc. All rights reserved.

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