SCHWEITZER 2002 Problem 7

Let the complex function $F(z)$ be regular on the punctuated disk $\{ 0<|z| < R\}$ . By a level curve we mean a component of the level set of $\mathrm{Re}F(z)$ , that is, a maximal connected set on which $\mathrm{Re}F(z)$ is constant. Denote by $A(r)$ the union of those level curves that are entirely contained in the punctuated disk $\{ 0<|z|<r\}$ . Prove that if the number of components of $A(r)$ has an upper bound independent of $r$ then $F(z)$ can only have a pole type singularity at $0$ .