If $\sum_{i=1}^{n}x_{i}=1$ and $x_{i}\ge 0$ then $\sum_{1\le i

Question

Show that $$\sum_{1\le i

for all $ n$-tuples $ (x_1, \ldots, x_n)$ satisfying $ x_i \geq 0$ and $ \sum_{i=1}^{n} x_i =1.$

I tried C-S, but without success.

Answer 1

We have $$ \sum_{1\le i

The bound is sharp (for $n \ge 2$) as can be seen by choosing $$ (x_1, x_2, x_3, \ldots, x_n) = (\frac 12, \frac 12, 0, \ldots, 0) \, . $$