Image of transcendental meromorphic functions

Question

Let $f$ be a trancendental meromorphic function such that $f'(z) \ne 0$ for all $z \in \mathbb{C}$. Let $\Pi$ be the stereoprojection map from the north pole on the unit sphere. My question is the following:

For any two points $P,Q \in \mathbb{C}$, can we find a curve $\gamma$ connecting $P$ and $Q$, such that $\Pi^{-1}(f(\gamma))$ lies in a great circle on the unit sphere, and that $\Pi^{-1}(f(\gamma))$ cover the circle at most once as points go from $P$ to $Q$ along the curve $\gamma$?

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Any ideas or comments are really appreciated!

Answer 1

No. A simple example is $f(z)=e^z$, $P=0$, $Q=10\pi i$. For any curve from $P$ to $Q$, the image is a closed curve which winds $5$ times around zero. So it cannot correspond to an arc of the great circle traversed once.