An inequality for Hausdorff means (Q2716468)

In this important and pleasant paper the author again delivers a new and interesting theme concerning elementary inequalities. He studies inequalities of the type NEWLINE\[NEWLINE\Biggl(\sum_n a_{m,n} \Biggl(\sum_k b_{n,k} x_k\Biggr)^p\Biggr)^{1/p}\leq \sum_n b_{m,n} \Biggl(\sum_k a_{n,k} x^p_k\Biggr)^{1/p}\quad (p\geq 1),\tag{\(*\)}NEWLINE\]NEWLINE wherein \(A\), \(B\) are prescribed matrices with nonnegative entries (e.g., Hausdorff means), and the estimate is to hold for all nonnegative sequences \({\mathbf x}\). Inequality \((*)\) holds not only in \(\ell^p\)-spaces, moreover its proof has nothing at all to do with \(\ell^p\)-properties, it is a consequence of the triangle inequality, and hence \((*)\) also holds for all norms.NEWLINENEWLINENEWLINEThe abstract and the introduction are very witty, but the many results are very essential, and mathematically exact.NEWLINENEWLINENEWLINEI warmly suggest to read entirely this nice work to everybody interested in elementary inequalities.