On the coefficient multipliers theorem of Hardy and Littlewood (Q1394494)

In the paper under review, the following result of Hardy and Littlewood, that is, for \(f(z)=\sum_{n=0}^{\infty}a_n(f)z^n\in H^p\), \(0<p<1\), there exists a constant \(C(p)\), dependent only on \(p\), such that \[ \sum_{n=0}^{\infty}\frac{|a_n(f)|^p}{(n+1)^{2-p}}\leq C(p)\big(\|f\|_p\big)^p, \] is sharpened by giving for \(C(p)\) the upper estimate \(C(p)\leq\pi e^p/[p(1-p)]\). Further, explicit estimates for sums \(\sum_{n=0}^{\infty}|a_n(f)\lambda_n|^s\), when \(f(z)=\sum_{n=0}^{\infty}a_n(f)z^n\) belongs to the mixed norm space \(H(1,s,\beta)\), provided \(0<s<\infty\), \(\beta>0\) and \[ \sup_{N\geq 0}\frac{\sum_{n=0}^N(n+1)^{s(1+\beta)}|\lambda_n|^s}{(N+1)^s}<\infty, \] are given. By using this technical result, the authors give a new version of some results by \textit{O. Blasco} [Can. J. Math. 47, 44-64 (1995; Zbl 0834.30032)] and by \textit{M. Jevtič} and \textit{M. Pavlovič} [Acta Sci. Math. 64, 531-545 (1998; Zbl 0926.46022)] in \(H(p,q,\alpha)\) for \(0<p\leq 1\), \(0<\alpha<\infty\) and \(0<q<\infty\).