On a result of Cartlidge
Let \(A=(a_{n,k})\) be a weighted mean matrix defined by \(a_{n,k}:={\lambda_k\over\Lambda_n}\) if \(1\leq k\leq n\) and \(a_{nk}:=0\) otherwise, where \(\lambda_1>0,\) \(\lambda_k\geq 0,\) and \(\Lambda_n:=\sum_{k=1}^n \lambda_k\). For any fixed \(1<p<\infty\), the author gives a simple proof of the following result. Theorem. If \(L:=\sup_n\Bigl(\frac{\Lambda_{n+1}}{\lambda_{n+1}}-\frac{\Lambda_n}{\lambda_n}\Bigr)<p\), then \[ \sum_{n=1}^\infty\,A_n^p\leq\bigl(\frac{p}{p-L}\bigr)\sum_{n=1}^\infty a_nA_n^{p-1}, \] where \(A_n:=\sum_ka_{nk}a_k\) (\((a_k)\in \ell^p\)). In particular, \(A\) maps \(\ell^p\) into \(\ell^p\) and the operator norm of \(A\) satisfies \(\| A\| _{p,p}\leq {p\over p-L}\). This result contains a result of \textit{J.\,P.\thinspace Cartlidge} [``Weighted mean matrices as operators on \(\ell^p\)'', Ph.\,D.\ thesis, Indiana University, 1978 (unpublished)] which generalizes Hardy's inequality [\textit{G.\,H.\thinspace Hardy, J.\,E.\thinspace Littlewood} and \textit{G.\,Pólya}, ``Inequalities'' (Cambridge University Press) (1988; Zbl 0634.26008)].
scientific article; zbMATH DE number 5161789