Finding Analytic Functions that Satisfy Certain Conditions

Question

How would one go about finding:

1. All analytic f such that $|f''(x)|>|e^{7z}|$?
2. All analytic functions such that $f(z) = z + f(z^2)$.
3. All linear functions that map from the punctured unit disk to itself.

Any hints would be much appreciated. I am not really sure how to give a general form as these questions ask.