Notes to G. Bennett's problems (Q1372379)

Let \(0<p,q<\infty\), \(s^{-1}= p^{-1}+ q^{-1}\), let \(\{a_n\}\) be a fixed sequence of nonnegative numbers with \(a_1>1\) so that \(A_n= \sum^n_{k= 1}a_k\) never vanishes, and let \(\{x_k\}\) be a sequence of real or complex numbers. \textit{G. Bennett} [Mem. Am. Math. Soc. 576, 130 p. (1966; Zbl 0857.26009)] proved that \[ \sigma_1:= \sum^\infty_{k=1} a_k\Biggl(\sum^\infty_{n= k}{| x_n|^s\over A_n}\Biggr)^{p/s}<\infty \] if and only if \[ \sigma_2:= \sum^\infty_{k= 1}| x_k|^s\Biggl({1\over A_k} \sum^k_{n= 1}| x_n|^s\Biggr)^{p/q}< \infty. \] Moreover, if \(p\leq q\), then (1) \(\sigma_1\leq ps^{-1}\sigma_2\), and if \(p\geq q\), then (2) \(\sigma_2\leq\sigma_1\). The aim of the paper is to prove that \(\sigma_1\leq K\sigma_2\) if \(p>q\), and \(\sigma_2\leq K\sigma_1\) if \(p<q\), provided that the sequences \(\{A_k\}\) and \(\{x_k\}\) satisfy some additional assumptions (\(K\) stands here for a positive constant).