Prove that $\left(x_1+\dots+x_k\right)^j\leq k^{j-1}\left(x_1^j+\dots+x_k^j\right)$ for $x_1,\dots,x_k\geq0$.

Question

Let $j,k\in\mathbb N$, with $j,k\geq 1$. Prove that $$\left(x_1+\dots+x_k\right)^j\leq k^{j-1}\left(x_1^j+\dots+x_k^j\right)$$ for $x_1,\dots,x_k\geq0$.

A proof by induction seems very difficult.

Answer 1

For $j \ge 1$ the map $x \mapsto x^j$ is convex on $\mathbb{R}^+$, so we can apply Jensen's inequality: $$\left(\sum \limits_{i = 1}^k x_i\right)^j = k^j \left(\sum \limits_{i = 1}^k \frac{1}{k} x_i\right)^j \le k^j \sum \limits_{i = 1}^k\frac{1}{k}x_i^j = k^{j - 1} \sum \limits_{i = 1}^k x_i^j$$