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Yes: by the generalized mean inequality (or, more specifically, by theAM--QM inequality), $\sqrt{nS}$ is an upper bound on $\sum_{k=1}^n\sqrt{s_k}$, which is better than
$n\sqrt{\max_{1\le k\le n}s_k}$.
Yes: by the generalized mean inequality, $\sqrt{nS}$ is an upper bound on $\sum_{k=1}^n\sqrt{s_k}$ better than
$n\sqrt{\max_{1\le k\le n}s_k}$.
Yes: by the generalized mean inequality (or, more specifically, by theAM--QM inequality), $\sqrt{nS}$ is an upper bound on $\sum_{k=1}^n\sqrt{s_k}$, which is better than
$n\sqrt{\max_{1\le k\le n}s_k}$.
