Prove inequality of following type other than in induction method

Question

I need to prove the following.

$$(1^r + 2^r +\ldots + n^r)^n > (n^n)(n!)^r$$

where $r$ being a real number.

I tried to solve it through induction method but it got complicated. How to solve this in other method?

Answer 1

Notice by $AM-GM$ inequality

$$ \frac{1^r + 2^r + ... + n^r}{n} \geq ( 1^r \cdot 2^r \cdot ... \cdot n^r)^{\frac{1}{n}} = ( (n!)^r)^{\frac{1}{n}}$$

Hence,

$$1^r + 2^r + ... + n^r \geq n (n!)^r)^{\frac{1}{n}} \iff (1^r + 2^r + ... + n^r)^n \geq n^n \cdot (n!)^n$$