Cauchy-Schwarz inequality for $a_1^4 + a_2^4 + \cdots + a_n^4 \geqslant n$

Also we can use Jensen. Let $f:\mathbb R \to \mathbb R, x\mapsto x^4$. Then $f$ is convex and thus by Jensen,

$$\frac{a_1^4+\dots+a_n^4}n=\frac{f(a_1)+\dots+f(a_n)}n\geq f\left(\frac{a_1+\dots+a_n}n\right)=\frac{(a_1+\dots+a_n)^4}{n^4}=1.$$