  We study one variable meromorphic functions mapping a planar real algebraic
set $A$ to another real algebraic set in the complex plane. By using the theory
of Schwarz reflection functions, we show that for certain $A$, these
meromorphic functions must be rational. In particular, when $A$ is the standard
unit circle, we obtain an one dimensional analog of Poincar\'e(1907),
Tanaka(1962) and Alexander(1974)'s rationality results for $2m-1$ dimensional
sphere in $\mathbb{C}^m$ when $m\ge 2$.
