Determine all the biholomorphic functions $\mathbb{C}\rightarrow \mathbb{C}$.
My attempt: First, we show that $z_{0}=0$ is not essential singularity of $g(z)=f\left(\frac{1}{z}\right)$, indeed, if $z_{0}=0$ was essential singularity of $g$, then for $\varepsilon>0$ we have $g\left(B_{\frac{1}{\varepsilon}}(0)\right)$ is dense in $\mathbb{C}$, or what it is the same, $f\left(\left\{z\in \mathbb{C} \left|\: \left|z\right|>\varepsilon \right. \right\}\right)$ is dense in $\mathbb{C}$.\ For other hand, as $f$ is biyective, then $f$ is not constant, then, by the open mapping theorem $f\left(\left\{z\in \mathbb{C} \left|\: \left|z\right|<\varepsilon \right. \right\}\right)$ is open in $\mathbb{C}$, and as $f\left(\left\{z\in \mathbb{C} \left|\: \left|z\right|>\varepsilon \right. \right\}\right)$ is dense in $\mathbb{C}$, then: $$f\left(\left\{z\in \mathbb{C} \left|\: \left|z\right|<\varepsilon \right. \right\}\right)\cap f\left(\left\{z\in \mathbb{C} \left|\: \left|z\right|>\varepsilon \right. \right\}\right)\neq \phi$$ but we know that $\left\{z\in \mathbb{C} \left|\: \left|z\right|<\varepsilon \right. \right\}\cap \left\{z\in \mathbb{C} \left|\: \left|z\right|>\varepsilon \right. \right\}= \phi$, so that $f$ is not inyective, which contradicts that $f$ is biyective.\ So that, $g(z)=f\left(\frac{1}{z}\right)$ has not essential singularity in $z_{0}=0$, therefore, by a lemma derived from Casorati–Weierstrass Theorem we have $=f\left(z\right)$ is a polynomial, therefore we have: $$f\left(z\right)=a_{0}+a_{1}z+\cdots +a_{m}z^{m} \qquad \mbox{for some } m\in \mathbb{C}.$$
My question: I think that to conclude the proof I must show that every function of the form $$f\left(z\right)=a_{0}+a_{1}z+\cdots +a_{m}z^{m}.$$ is biholomorphic, but for me it is not clear.
The lemma you want to use is probably the following
Lemma : If $f$ is holomorphic and injective on $\mathbb{C}\backslash\{z_0\}$ and $z_0$ is not removable, then $z_0$ is a simple pole.
This allow you to state that $a_m =0$ if $m>1$. (Because $f(\frac{1}{z})$ is holomorphic and injective on $\mathbb{C}_0$.) Hence the biholomorphic functions on $\mathbb{C}$ have the form $f(z) = az+b.$