Interesting inequality question

Question

There is one interesting question in my homework that has very elegant form but also gives me hard time. Not sure if this question is based on any theorem or not, but I would like to know if it is. Anyhow, here is the question

If $0<\alpha<1$, and $n \in \mathbb{N}$ show that

$\sum_{i=1}^n a_i^{\alpha} \gt \left( \sum_{i=1}^n a_i\right)^\alpha$ for any $a_1,a_2,\ldots a_n \gt 0$

So, I need some ideas/hints on how to tackle this question, not straight-forward answer. Thanks for help.

Answer 1

Hint:

Use induction to reduce to the case $n=2$.

Answer 2

Both of sums are equal to zero when all $a_i$ are zeros.

Using derivative you can find out that the first sum grows at least not slower than the second with growth of any $a_i$.