A short note on weighted mean matrices (Q1306813)

The authors show that, if \(\{p_n\}\) and \(\{q_n\}\) are two positive sequences satisfying the conditions \(P_n(q_{n+1}/p_{n+1}-q_n/p_n) \approx(n+1)^\alpha\) for some \(\alpha\geq 0\), \(P_nq_n/p_nQ_n\leq H\), \(H\) a constant, and \(\{p_n\}\) is nondecreasing, then \((N,p_n)\) is a bounded operator on \(\ell^p\) whenever \((N,q_n)\) is a bounded operator on \(\ell^p\), \(1<p<\infty\). This result generalizes a theorem of the reviewer [Indian J. Math. 36, No. 1, 1-6 (1994; Zbl 0864.40003)].