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 \(\{\mu_k\}_{k=1}^{\infty}\).


26D15 Inequalities for sums, series and integrals
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