I know this isn't a very typical question, but i was wondering if anyone knows any good lower bounds for $a^n + b^n$. I'm looking for something akin to $a^n + b^n \leq (a + b)^n$ for $n \geq 1$.
The motivation is that I'm trying to find some nice upper bounds on $N$ for the minimum value of $N$ when $a_1^N + a_2^N + \dots + a_k^N \geq M$, for some given $1 \leq a_1, \dots, a_k \leq 2$ and $M > 0$. I don't need this to be particularly tight, but I was thinking if I could write this as some expression $a_1^N + a_2^N + \dots + a_k^N \geq f(a_1, a_2, \dots, a_k)^N \geq M$, then if $f$ is nice enough the $\log$ upper bound on $N$ should be good enough for my purposes.
I've tried using the AM-GM to get $a^n + b^n \geq nab$, but as that doesn't give a logarthmic bound it's not good enough for my purposes. I've scrolled through a bunch of other lists of inequalities and I can't find anything else that seems to work.
Does anyone have any ideas? Either for my original question or for the motivation? Thanks for the help!