Lower bounds for generalized Hausdorff matrices and lower triangular matrices on the block weighted sequence space lp(w, F)

Let 0 < p < 1 and H = (h(n),(k))(n, k) (>=) (1) be a non-negative matrix. Denote by L-w,L- (p,) (q,) (F) (H), the supremum of those L, satisfying the following inequality {Sigma(infinity)(n=1)wn (Sigma(j is an element of Fn) Sigma(infinity)(k=1)h(j,kxk))(q)}(1/q) >= L(Sigma(infinity)(n=1)w(n){Sigma(j is an element of ln)x(j)}(p))(1/p), where F = (F-n) is a partition of positive integers, I-n = {n}, x is a non-negative sequence in l(p)(w, I) and (w(n)) is a monotone and non-negative sequence of real number. In this paper a Hardy-type formula is obtained for L-w,L- (p,) (q,) (F) (H-mu(alpha)), where H-mu(alpha) is the generalized Hausdorff matrix, 0 < q <= p < 1 and alpha > 0. Another purpose of this paper is to establish a general upper estimate for the exact value of L-w,L- p,L- I (H-t), for which recently a lower estimate was established in Lashkaripour and Talebi [Lashkaripour R, Talebi G. Bull. Iran. Math. Soc. 2011;37:115-126], where H is a non-negative lower triangular matrix and 0 < p < 1. We also derive the corresponding result for L-w,L- p,L- I (H), with -infinity < p < 0. In particular, we apply our results to summability matrices, weighted mean matrices, Norlund matrices. Our results also generalize some results in Chen and Wang [Chen C-P, Wang K-Z. Linear Multilinear Algebra, March 2011;59:321-337] and Lashkaripour and Talebi [Lashkaripour R, Talebi G. Czech. Math. J. 2012; 62: 293-04.].