I am stuck at a step in a problem where I've been given hints. The hints confuse me, so I'm hoping for some help.
Assuming that $f$ is entire and that $$|f(z)| \leq |z| + 1/|z|, $$ for $z \in \mathbb{C} \backslash \{0\},$ I wish to show that $f(z)$ is of the form $az+b$.
I want to use the taylor expansion of $f$, so that $$f(z) = \sum_{n=0}^{\infty} a_n z^n. $$
The hints say to use the Cauchy estimates and the maximum principle to show that $$|a_n|r^n \leq \max_{|z|=r}|f(z)|\leq r + 1/r, $$
so that as $r \rightarrow \infty$, we get $a_n = 0$ for $n \geq 2$.
It is the first inequality that confuses me. The maximum principle says that if $|f(z)| \leq M$ for $|z| = r$, then $|f(z)| \leq M$ for $|z| \leq r$. But I don't see how that is even relevant in this case. And I've had even less luck using the Cauchy estimates.