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Notes to G. Bennett’s problems. (English) Zbl 0894.26008
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.
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).
Reviewer: B.Opic (Praha)
##### MSC:
 26D15 Inequalities for sums, series and integrals 40A05 Convergence and divergence of series and sequences
##### Keywords:
infinite sums; inequalities; quasi-monotone sequences
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