Let $V$ be a Riemann surface, $x\in V$, and $B:=B(x,r)$ some small ball (in a local chart). It is well known that there is a meromorphic function $f$ on $V$ with the only pole at $x$. What I’d like to ask of is if there is a meromorphic f on V that has a pole at x and additionally such that $|f|<1$ outside $B$?



Meromorphic function on the Riemann surfaces
For open surfaces, there are counterexamples. The first one was constructed by P. Myrberg:


Ueber die analytische Fortsetzung von beschrankten Funktionen, Ann. Acad. Sci. Fenn., Ser. A. I N:o 58 (1949)


Since this paper is difficult to obtain (and written in German), I refer to another paper


Heins, Maurice,
Riemann surfaces of infinite genus.
Ann. of Math. (2) 55 (1952), 296–317,


Which proves an even stronger result: there is an open Riemann surface, such that if you remove a disk from it, then on the remaining surface every non-constant meromorphic function takes all complex values, except at most two of them.


