Let \(\displaystyle f(z)\) be a meromorphic function on the complex plane, and suppose there is an integer \(\displaystyle m\) such that \(\displaystyle f^{-1}(w)\) has at most \(\displaystyle m\) points for all \(\displaystyle w \in \mathbb{C}\). Show that \(\displaystyle f(z)\) is a rational function.
The back of the book says:
Choose \(\displaystyle w_0\) such that the number of points in \(\displaystyle f^{-1}(w_0)\) is maximum. Then \(\displaystyle f(z)\) attains values \(\displaystyle w\) near \(\displaystyle w_0\) only near points in \(\displaystyle f^{-1}(w_0)\), \(\displaystyle \frac{1}{f(z)-w_0}\) is bounded at \(\displaystyle \infty\), and \(\displaystyle f(z)\) is meromorphic on \(\displaystyle \overline{\mathbb{C}}\) hence rational. This seems a bit too sketchy. I think that \(\displaystyle \frac{1}{f(z)-w_0}\) is bounded at \(\displaystyle \infty\) by taking \(\displaystyle \lim_{z \rightarrow \infty} f(z)\). How is \(\displaystyle f(z)\) meromorphic on \(\displaystyle \overline{\mathbb{C}}\)? Also, how does \(\displaystyle f(z)\) attain values \(\displaystyle w\) near \(\displaystyle w_0\) only near points in \(\displaystyle f^{-1}(w_0)\). I understand that it is the maximum. However, I don't see precisely why. Thanks.
The back of the book says:
Choose \(\displaystyle w_0\) such that the number of points in \(\displaystyle f^{-1}(w_0)\) is maximum. Then \(\displaystyle f(z)\) attains values \(\displaystyle w\) near \(\displaystyle w_0\) only near points in \(\displaystyle f^{-1}(w_0)\), \(\displaystyle \frac{1}{f(z)-w_0}\) is bounded at \(\displaystyle \infty\), and \(\displaystyle f(z)\) is meromorphic on \(\displaystyle \overline{\mathbb{C}}\) hence rational. This seems a bit too sketchy. I think that \(\displaystyle \frac{1}{f(z)-w_0}\) is bounded at \(\displaystyle \infty\) by taking \(\displaystyle \lim_{z \rightarrow \infty} f(z)\). How is \(\displaystyle f(z)\) meromorphic on \(\displaystyle \overline{\mathbb{C}}\)? Also, how does \(\displaystyle f(z)\) attain values \(\displaystyle w\) near \(\displaystyle w_0\) only near points in \(\displaystyle f^{-1}(w_0)\). I understand that it is the maximum. However, I don't see precisely why. Thanks.