Let $ b > 1$. Let $f(z)$ be -meromorphic over the entire complex plane. Also $f(z)$ maps the real line to the extended real line. And $f(z)$ is non-linear.
Consider the strip $A$ such that $ 0 =< \Im(z) < \frac{2 \pi}{\ln(b)}$.
1) Is it true that there is always a complex number $q$ in $A$ such that $f(\exp(b q)) = q$ ?
2) Consider $ f(\exp(b s)) = 0 $ such that $s$ is in $A$. Is it true that when $f$ is a rational function, you can get zero's at infinity AND for finite Numbers at the same time ?