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On an inequality in \(l_p(\mathbb C)\) involving Basel problem. (English) Zbl 1337.26043

The authors study the following conjecture: For all \(a=(a_{1}, a_{2}, \ldots )\in l_{2}(\mathbb C)\), the inequality \[ \sum_{n\geq 1}|a_{n}|^{2} \geq \frac{6}{\pi^{2}}\sum_{k\geq0}\left|\sum_{l\geq 0}\frac{1}{l+1}a_{2^{k}.(2l+1)}\right|^{2} \] posed by Z. Retkes [http://www.openproblemgarden.org/category/analysis] holds, and they give a very simple proof. In addition, an extension of the above conjecture to \(l_{p}(\mathbb C)\), \(p>1\) is also given.

MSC:

26D15 Inequalities for sums, series and integrals
26D20 Other analytical inequalities
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