Summary: Let \(A=(a_{n,k})_{n,k\geq 1}\) be a matrix operator, \((v_n)^\infty_{n=1}\) and \((w_n)^\infty_{n=1}\) be two non-negative sequences and \(p\geq 1\). This paper is focused on the problem of finding the infimum of those \(U\), satisfying the following inequality: \[ \left( \sum^\infty_{n=1} w_n\left| \frac{1}{n} \sum^\infty_{k=1} \sum^n_{i=1} a_{i,k} x_k \right|^p\right)^{\frac{1}{p}} \leq U \left( \sum^\infty_{n=1}v_n |x_n|^p\right)^{\frac{1}{p}}, \] for all sequences \(x\in l_p\). We apply our results to well-known matrix operators such as backward difference, quasi-summability, weighted mean, Hilbert and Nörlund.