Computing modulus of complex function?

Question

Question: Suppose that $f$ is an entire function and that $|f(z) − z + 2z^2| ≤ \arctan(|z|)$ for all $z ∈ \Bbb C$. Compute $f(2)$.

I have been trying to think about how the "entire" property would help me, but I couldn't find any theorems nor definitions to apply. Should I just unwrap the function just like the real functions since this is involves absolute value and inequalities?

I am using the Stein and Shakarchi's Complex Analysis, Chapter 5 was for entire function.

Any help would be greatly appreciated! :)

Answer 1

Hints:

1. What is $f(0)$?
2. $\arctan{|z|}$ is bounded. What does this tell you about the left-hand side?