An inequality about non negative numbers Question:
Let $x_1, x_2, ... , x_m$ be non negative numbers, and let $\sum_{i=1}^{m}x_i=k.$ If $s>1,$ then
$$\sum_{i=1}^{m}x_i^s \geq \frac{k^s}{m^{s-1}}.$$ Equality holds iff $x_i=k/m, i=1 , 2 , ... , m.$
I am trying to prove the given inequality using induction or A.M and G.M inequalities but am not able to prove it.
Any hint or solution is appreciated
Thank you!


Jensen's inequality:
If $\lambda_1,$ $\lambda_2,$ $...,$ $\lambda_n$ be non negative real numbers such that $\lambda_1+\lambda_2+...+\lambda_n=1,$ and $f$ is convex function then $f(\lambda_1x_1+\lambda_2x_2+...+\lambda_nx_n) \leq \lambda_1f(x_1)+\lambda_nf(x_n)+...+\lambda_nf(x_n),$ for any $x_1,x_2,...,x_n.$

Let $\phi :$ $x \to x^s$ be a convex function for $x>0$ and $s>1$ then,
By Jensen's inequality
$\phi((1/k)x_1+(1/k)x_2+...+(1/k)x_n) \leq (1/k) \phi(x_1)+(1/k) \phi(x_2)+...+(1/k) \phi(x_n)$
$\implies ((1/k)x_1+(1/k)x_2+...+(1/k)x_n)^s \leq (1/k)x_1^s+(1/k)x_2^s+...(1/k)x_n^s$
$\implies (1/k)^s(x_1+x_2+...+x_n)^s \leq  (1/k)(x_1^s+x_2^s+...+x_n^s)$
$\implies (x_1^s+x_2^s+...+x_n^s) \geq \frac{(x_1+x_2+...+x_n)^s}{k^{s-1}}.$
This is my approach to proving the given inequality using the convex function and Jensen's inequality.
