Does this funcion define a norm on $\mathbb{C}^n$?

Question

Let $m$ and $n$ be two given positive integers. And, let $f \colon \mathbb{C}^n \to \mathbb{R}$ be defined as follows: $$ f(x_1, x_2, \ldots, x_n) \colon= \left( \sum_{i=1}^n \sqrt[m]{|x_i|} \right)^m$$ for all $\vec{x} \colon= (x_1, x_2, \ldots, x_n) \in \mathbb{C}^n$.

Then does $f$ define a norm on $\mathbb{C}^n$?

N1: $f$ is non-negative.

N2: If $f(\vec{x}) = 0$, then $\vec{x} = \vec{0}$.

N3: $f(\alpha \vec{x}) = |\alpha| f(\vec{x})$.

What about the triangle inequality?

Answer 1

Hint: Try to consider the easy case: $m=n=2$.

In general case you can show that you obtain the inverse triangle inequality, i.e. $f(a+b)\ge f(a)+f(b)$.