Let 1 < p < ∞ and A = (a n,k) n,k 1 be a non-negative matrix. Denote by A w,p,F , the infimum of those U satisfying the following inequality: Ax w,p,F U x w,p,I , where x 0 and x ∈ l p (w,I) and also w = (w n) ∞ n=1 is a decreasing, non-negative sequence of real numbers. The purpose of this paper is to give a lower bound for A w,p,F , where A is a lower triangular matrix. In particular, we apply our results to Weighted mean matrices and Nörlund matrices which recently considered in [2,3,6] on the usual sequence spaces. Our results generalize some work of Jameson, Lashkaripour, Frotannia and Chen in [4,7,8].