Consider the analytic function $f : (0,\infty) \to (0,\infty)$ given by
$$
f(x) = \bigg( \sum\_{i=1}^n a\_i b\_i^{1/x} \bigg)^x
$$
where $n\in\mathbb N$, $a\_i>0$ and $0<b\_1<\ldots<b\_n<1$. I am interested in analytic continuations of $f$ to the right half plane $\mathbb H\_+ = \{z\in\mathbb C : \mathrm{Re}(z)>0\}$.


My guess is that that such an analytic continuation to all of $\mathbb H\_+$ exists only in the trivial case $n=1$ (where the continuation is $f(z) = b\cdot a^{z}$). Even the $n=2$ case seems to be quite hard.


Any suggestions and ideas would be much appreciated!



Analytic continuation of a function on the half line
Partial answer:


$g(z)=\sum\_{k=1}^n a\_kb\_k^{1/z}$ is analytic on the punctured plane and clearly $f(z)=e^{z\log g(z)}$ defines $f$ analytic outside the zeroes of $g$ - as usual locally and then we can stitch together a global $f$ with appropriate cuts.


Now $g >0$ on the real axis outside the origin by definition, so it comes down to figuring out if $g$ has zeroes only in the left-hand plane.


Since $1/z$ preserves the right hand and left hand planes, we need to look at the zeroes of $\sum a\_kb\_k^w=\sum a\_ke^{c\_k w}, c\_k=\log b\_k$ which are known as exponential or Ritt polynomials and there is a large literature about.


Edit later - actually since in the case $n=2$ the zeroes of $h(w)=a\_1e^{c\_1w}+a\_2e^{c\_2w}, c\_k=\log b\_k, c\_1<c\_2<0$ are clearly periodic with period $\frac{2k \pi i}{c\_2- c\_1}$, so all have the same real part, the function $f$ will be analytic in the right half plane precisely when one, hence all, are in the left half plane as then the zeroes of $g(z)=h(1/z)$ are like that too as noted.


This happens when with $a=\frac{a\_1}{a\_2}$ so one such zero is $w\_0=\frac{\log (-a)}{c\_2-c\_1}$ we have $\Re w\_0 \le 0$ and that happens precisely when $|a| \le 1$ or $a\_1 \le a\_2$, so we can actually completely solve the case $n=2$


Edit later per comments: Since $f(x)$ seems to be defined the usual way for $x>0$ where we have a unique positive $1/x$ root for the positive number $g(x)$, $\log g(x)$ is defined uniquely and continuously (real analytically too) by its usual real value and we can extend it (uniquely) to an analytic $\log g$ on a small (simply connected) neighborhood of the positive real axis in $\Re z>0$ - note that it may not be a strip if zeroes of $g$ accumulate towards the real axis. So already $f$ has unique analytic continuation to a complex open set in $\Re z >0$ and the only question is how much more we can go and still be analytic in $\Re z >0$ and that is precisely until we hit zeroes of $g$ where we have to start making cuts. If there are no zeroes, then $\log g$ and hence $f$ have analytic continuation to the whole half plane.


If $n=1$ we have $g(x)=a^xb, x>0$ so $\log g(x)=x \log a +b, x>0$ and that clearly extends to $\Re z>0$ so $f(z)=b a^z$ is the analytic continuation to $\Re z >0$ and actually to all $z$ finite as the singularity at $0$ is removable as we see.


