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?

Another way to look at it is to use Titu's Lemma which is also known as Cauchy-Schwarz inequality in Engel form.
$$\sum_{i=1}^n\frac{\alpha_i^2}{1}\geq \frac{(\sum_{i=1}^n\alpha_i)^2}{\sum_{i=1}^n1}=\frac{(\alpha_1+\alpha_2+\ldots+\alpha_n)^2}{n}\\ \implies n({\alpha_1^2 + … + \alpha_n^2})\geq (\alpha_1 + … + \alpha_n)^2\\ \implies (\alpha_1 + … + \alpha_n)^2\leq n({\alpha_1^2 + … + \alpha_n^2})$$
Equality holds iff $a_1=a_2=\ldots=a_n$
