# Image of transcendental meromorphic functions

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 $$\gamma$$?

Any ideas or comments are really appreciated!

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.
• By the way, you mean $P=0$, right? Jun 3, 2020 at 14:57