For which values of $p,q\in[1,\infty)$ the following functions have a real analytic continuation to the whole real line.
1.$f:(0,\infty)\to \mathbb R$ where $f(t)=(1+t^p)^{\frac{q}{p}}$.
2.$g:(-\infty,0)\to \mathbb R$ where $g(t)=(1+|t|^p)^{\frac{q}{p}}$. The above functions are actually $\ell_p$ norm of $(1,t)\in(\mathbb R^2.\|.\|_p)$ when $q=1.$ that is why I want to know if they can be extended analytically. This is my way to understand the $\ell_p$-norms in a better way.
For 1, the answer is that $p \ge 1$ needs to be an integer and if it is odd then $q/p$ must be an integer too so $f$ is a polynomial; if $p$ is even $q$ can be any complex number for that matter; for 2, $g(t)=f(-t)$ so it is essentially same problem
I will only sketch the proof as there are only two properties we use, namely derivatives of analytic functions are analytic and analytic continuation is unique -
assume $p$, not an integer, then $f'(t)=t^{p-1}q(t)$ and if we keep differentiating we eventually get to some $f^{(k)}(t)=t^{p-k}q_k(t)+r_k(t), p-k <0$ minimal, so $q_k, r_k$ are continuos at zero; this shows that $f^{(k)}(t) \to \infty, t \to 0, t>0$ so $f$ cannot be continued even as a $C^k$ function at $0$
If $p$ is even then $1+t^p>0, t \in \mathbb R$ so we can define an analytic logarithm $\log (1+t^p)$ on a simply connected neighborhood of $\mathbb R$ in $\mathbb C$ (eg narrow strip) which coincides with the usual logarithm on positive numbers, and then $f(z)=\exp ((q/p)(\log(1+z^p)$) is an analytic continuation of the original $f$ to that strip for any $q$ and we can take $q$ complex for that matter too
If $p$ is odd, then $1+t^p>0, t > -1$ so we can do as above and get an analytic continuation to $(-1, \infty)$ that is $f(t)=\exp ((q/p)(\log(1+t^p)$) and is unique as noted; but now if $q/p$ is not a positive integer, taking enough derivatives we will again get that $|f^{(k)}(t)| \to \infty, t \to -1, t> -1$ so we cannot continue the function beyond $-1$ and we are done!