A note on absolute Nörlund summability factors (Q1803961)

The author proves the Theorem: Let \(p_ 0>0\), \(p_ n\geq 0\) and let \((p_ n)\) be a nonincreasing sequence and let \(k\geq 1\). If \(\sum^ n_{v=1}{1\over v}| t_ v|^ k=O(X_ n)\) as \(n\to\infty\), and the sequence \((X_ n)\) and \((l_ n)\) are such that (i) \(\sum^ \infty_{n=1}nX_ n|\Delta^ 2l_ n|<\infty\) and (ii) \(l_ nX_ n=O(1)\) as \(n\to\infty\). Then the series \(\sum a_ nl_ nP_ n(n+1)^{-1}\) is summable \(| N,p_ n|_ k\). This result is a generalization of his previous result proved for \(| N,p_ n|\).