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Inequality on sum

Source: IMC 2000 day 1 problem 4

October 29, 2005
inequalitiesreal analysisreal analysis unsolved

Problem Statement

Let (xi)(x_i) be a decreasing sequence of positive reals, then show that: (a) for every positive integer nn we have ∑i=1nxi2≤∑i=1nxii\sqrt{\sum^n_{i=1}{x_i^2}} \leq \sum^n_{i=1}\frac{x_i}{\sqrt{i}}. (b) there is a constant C for which we have ∑k=1∞1k∑i=k∞xi2≤C∑i=1∞xi\sum^{\infty}_{k=1}\frac{1}{\sqrt{k}}\sqrt{\sum^{\infty}_{i=k}x_i^2} \le C\sum^{\infty}_{i=1}x_i.
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