Liouville and another beautiful year

Calculus Level 4

Suppose $$f: \mathbb{C} \to \mathbb{C}$$ is a holomorphic function. Furthermore, assume $|f(z)| \le 2016|z|$ for all $$z\in \mathbb{C}$$.

If $$f(2016) = 6048 + 8064i$$, what is $$f(3 - 4i)$$?


Note: $$i = \sqrt{-1}$$

Inspiration

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