On Hardy's inequality
The following result is proved: Let \(0< b_{n+1}\leq b_n\), \(B_n= \sum^n_{k= 1}b_k\), \(a_n\geq 0\), \(0< \sum^\infty_{n=1} b_na_n< \infty\). Then \[ \sum^\infty_{n= 1} b_{n+ 1}(a^{b_1}_1\cdots a^{b_n}_n)^{1/B_n}< e \sum^\infty_{n= 1} \Biggl[1- {b_n\over 2(B_n+ b_n)}\Biggr] b_na_n. \] For \(b_n\equiv 1\) a refinement of Carleman's inequality follows. The proof uses Pólya's method [see, e.g., \textit{G. H. Hardy}, \textit{J. E. Littlewood} and \textit{G. Pólya}: ``Inequalities'' (1952; Zbl 0010.10703; 2nd ed. 1988; Zbl 0634.26008)] as well as certain inequalities for the number \(e\). Lemma 2 is well known, even in improved forms [see, e.g., \textit{J. Sándor}, Elem. Math. 43, No. 6, 177-180 (1988; Zbl 0702.26016)].
scientific article; zbMATH DE number 1334734