In his book Serge Lang provides a discussion and then proposing the Open Mapping Theorem as being obvious in the light of the discussion but I am not too sure if I am getting it correctly.
My approach:
Due to the convergence of $f(z)$ there exists A, sufficiently large, such that $a_n\leq A^n$. Then define $F(z)=A^mz^m(1+h(z))$ . We note $f(z) \leq F(z)$ and then we proceed as the discussion ariving at $F(z)=F(z)^m$ and that $F$ is an open mapping. But $f(z) \leq F(z))$, so in the case of $f(z)<F(z)$, the function $f(z)$ will be bounded (or atleast, since I can't prove its open).
Can somebody please explain to me what Serge Lagn's implied proof is for the general $f(z)=\sum a_nz^n$ with $a_n$ not neccesarly beeing the same as $a^n$?
What Serge Lang did is taking $n$-root. Let me clarify his argument. To prove that $f$ is local surjective in a neighborhood of $a$, he used the power series expansion of $f$ $$f(z)=f(a)+C(z-a)^n + \ldots$$ where $C$ is a nonzero constant. Hence $$f(z)=f(a)+(z-a)^ng(z)$$ with $G(a)\neq 0$. There exists an open disc $U$ centered at $a$ that $g$ is nonzero, so $g$ admits a $n$-root $g_1$, i.e. $g(z)=g_1^n(z)$ on $U$. Note that $g_1(a)\neq 0$. Rewrite $f$ we have $$f(z)=f(a)+[(z-a)g_1(z)]^n=f(a)+u^n(z)$$ The map $u:z\mapsto (z-a)g_1(z)$ is local analytic isomorphism by Theorem 6.1(c) above, so we can take an open neighborhood $V\subset U$ of $a$ that $u$ maps it isomorphically to an open disc $D$ centered at $0$. The map $w\mapsto w^n$ maps $D$ onto another open disc, hence the image of $f$ contains an open disc of $f(a)$.
In his proof, he didn't want to construct general $n$-root yet, so he did it formally using power series expansion. Indeed, the binomial expansion he used is $$\begin{aligned} (1+h(z))^{1/m} &=1+\frac{1}{m}h(z)+\frac{1}{2}\frac{1}{m}(\frac{1}{m}-1)h^2(z)+\ldots\\ &=1+h_1(z) \end{aligned}$$ I don't know how he prove the uniform convergent of the above series, but the uniqueness of power series expansion show that if $h_1$ exists, it must satisfy the equations above.
For the construction of logarithm and $n$-root, if Lang didn't treat them in his book, you should look at other books. The proof of Open Mapping Theorem usually follows straightforwardly from several important results (often Rouché's Theorem). I haven't checked out Lang's book yet, but I think he postponed the treatment of logarithm and $n$-root until he introduced Riemann surfaces, maybe?