Inequality for complex numbers

[vale edge]

Is the following inequality true for complex numbers x_i? There is no need to prove formally, I wish to understand intuitively why this inequality holds (or doesn't)

(sum(|x_i|))^1/2 <= sum(|x_i|^1/2)

where sum(x_i) is: x_1 + x_2 + ... + x_k

[outer shore]

And x_1, x_2, etc. are all complex numbers?

[muted inlet]

$\bigg(\sum_{i=1} |x_i|\bigg)^{\frac{1}{2}} \leq \sum_{i=1} |x_i|^{\frac{1}{2}}\$

where $\sum_{i=1} x_i = x_1 + x_2 + ... + x_k$

[blissful acorn]

adonhs

[hollow dragon]

using the fact that sqrt is concave you can show an even stronger inequality

@vale edge

now as for intuition

¯_(ツ)_/¯

it's not like it's even specific to modules of complex numbers or anything

replace your |x_i| and just have x_i positive real numbers instead and it still works fine

[vale edge]

Yes

[vale edge]

Hmm ok. Thinking this way is clearer. I imagine using indution is a way to go, too