I apologize for the poor title.
Let $p\in 2\mathbb N$ and $a_1,\ldots , a_n\geq 0$.
Show that $$\left (\sum a_j\right )^p \leq n^{p-1}\sum a_j^p $$
So, firstly, by rearranging the expression, it's sufficient to show that $$\left (\sum\frac{a_j}{n}\right )^p \leq \sum\frac{a_j^p}{n}. $$ It's known from theory of integrals that for any convex function $g:\mathbb R\to\mathbb R$ $$g(EX)\leq E g(X) \quad\mbox{the Jensen inequality}$$ Suppose $(\Omega,\mathcal F,\mathbb P)$ is a probability space with $X_1,\ldots, X_n$ some $\mathcal F$-measurable functions such that $EX_j \equiv a_j$. Since $p\in2\mathbb N$, among others, the mapping $x\mapsto x^p$ is convex. Define $$X := \sum \frac{X_j}{n}, $$ The inequality yields $$g(EX) = g\left (\int\sum\frac{X_j}{n}\mbox{d}\mathbb P\right ) = \left (\sum\frac{a_j}{n}\right )^p\leq \int\left (\sum\frac{X_j}{n}\right )^p\mbox{d}\mathbb P = Eg(X) $$ ..and I don't see any way to make progress or I'm missing something very trivial or is the choice of $g$ or $X$ a poor one? Are there any suggestions?