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Some reverse $$l_{p}$$-type inequalities involving certain quasi monotone sequences. (English) Zbl 1355.26032
A monotonically decreasing real sequence $$\{\lambda_k\}_{k=1}^{\infty}$$ is called lacunary monotone if there exist positive constants $$r_1$$ and $$r_2$$ such that, for every $$k \in \mathbb{N}$$, one has $$r_1 \lambda_{2^k} \leq \lambda_{2^{k+1}} \leq r_1 \lambda_{2^k}$$. Let $$\{a_k\}_{k=1}^{\infty}$$ be a monotonically decreasing sequence. Suppose that $$\{\lambda_k\}_{k=1}^{\infty}$$ and $$\{\mu_k\}_{k=1}^{\infty}$$ are lacunary sequences. Let $$m, n$$ and $$p$$ be positive integers such that $$m \geq 16 n$$ and $$p \geq 1$$. One of the main results of the authors is in showing that the following inequality holds: $\sum_{l=n}^m \lambda_l (\sum_{k=1}^l a_k \mu_k)^p \geq \alpha \sum_{l=4n}^m \lambda_l (la_l\mu_l)^p,$ where the constant $$\alpha$$ depends only on $$p$$ and the sequences $$\{\lambda_k\}_{k=1}^{\infty}$$ and $$\{\mu_k\}_{k=1}^{\infty}$$.

MSC:
 26D15 Inequalities for sums, series and integrals
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