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Apply Jensen's inequality (Theorem T1.1) to the convex function \(\varphi(x) = x^2\) with equal weights \(\lambda_k = 1/n\) to derive the mean-square inequality: for every \(x_1, \dots, x_n \in \mathbb{R}\), $$ \left( \frac{x_1 + \dots + x_n}{n} \right)^2 \le \frac{x_1^2 + \dots + x_n^2}{n}. $$
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