Authors’ abstract: Let \(0<p<1\) and \(H=\left( h_{n,k}\right) _{n,k\geq 1}\) be a non-negative matrix. Denote by \(L_{w,p,q,F}\left( H\right) \), the supremum of those \(L\), satisfying the following inequality \[ \left\{ \sum\limits_{n=1}^{\infty }w_{n}\left( \sum\limits_{j\in F_{n}}\sum\limits_{k=1}^{\infty }h_{j,k}x_{k}\right) ^{q}\right\} ^{1/q}\geq L\left\{ \sum\limits_{n=1}^{\infty }w_{n}\left( \sum\limits_{j\in I_{n}}x_{j}\right) ^{p}\right\} ^{1/p}, \] where \(F=\left( F_{n}\right) \) is a partition of positive integers\(,I_{n}=\left( n\right) \), \(x\) is a non-negative sequence in \(l_{p}\left( w,I\right) \) and \(\left( w_{n}\right) \) is a monotone and non-negative sequence of real number. In this paper, a Hardy-type formula is obtained for \(L_{w,p,q,F}\left( H_{\mu }^{\alpha }\right) \), where \(H_{\mu }^{\alpha }\) is the generalized Hausdorff matrix, \(0<q\leq p<1\) and \(\alpha >0\). Another purpose of this paper is to establish a general upper estimate for the exact value of \( L_{w,p,I}\left( H^{t}\right) \), for which a lower estimate was established recently in [R. Lashkaripour and G. Talebi, Bull. Iran. Math. Soc. 37, No. 1, 115–126 (2011; Zbl 1242.40004)], where \(H\) is a non-negative lower triangular matrix and \(0<p<1\). We also derive the corresponding result for \( L_{w,p,I}\left( H\right) \), with \(-\infty <p<0\). In particular, we apply our results to summability matrices, weighted mean matrices, Nörlund matrices. Our results also generalize some results in [C.-P. Chen and K.-Z. Wang, Linear Multilinear Algebra 59, No. 1–3, 321–337 (2011; Zbl 1229.47047)] and [R. Lashkaripour and G. Talebi, Czech. Math. J. 62, No. 2, 293–304 (2012; Zbl 1265.26074)].