Absolute summability factors (Q1110758)

\textit{S. N. Lal} [Proc. Am. Math. Soc. 14, 311-319 (1963; Zbl 0117.294)] proved the following: Theorem A: Let the sequence \((y_ n)\) be (1) \(\sum^{\infty}_{n=1}n^{-1}| y_ n| <\infty\). Then (2) \(f\in S(f;\phi_ 1)\Rightarrow \sum^{\infty}_{n=1}A_ n(x)\epsilon_ n\in | N,1/(n+1)|\) and Theorem B: Let \((y_ n)\) satisfy (1) and (2) and let \(\sum^{n}_{m=1}md_ m=O(n).\) Then \(\sum^{\infty}_{n=1}d_ n\epsilon_ n\in | N,1/(n+1)|\) we first observe that if \((y_ n)\) satisfies (1) and (2), then \(0\leq \Delta y_ n\downarrow\), \(\sum^{\infty}_{n=1}\log (n+1)\Delta y_ n<\infty.\) The author improves the theorems of Lal, replacing condition (1) (of theorem A) by \(\sum^{\infty}_{n=1}n^{-1}| \Delta y_ n| \log (n+1)<\infty\) and proved that the summability method \(| R,e(n),1|\) is weaker than \(| N,1/(n+1)|\).