Prove $(\alpha_1 + ........ + \alpha_n)^2 ≤ n \cdot (\alpha_1^2 + ....... + \alpha_n^2)$ For any real numbers $\alpha_1, \alpha_2, . . . . ., \alpha_n$,
$$(\alpha_1 + ...... + \alpha_n)^2 ≤ n \cdot (\alpha_1^2 + ..... + \alpha_n^2)$$
And when is the inequality strict?

This is just an application of Jensen's inequality. As $f(x) = x^2$ is a convex function, we can apply it here.
Reference : Jensen's inequality
