Find all analytic functions such that... Here is the problem: find all functions that are everywhere analytic, have a zero of order two in $z=0$, satisfy the condition $|f'(z)|\leq 6|z|$ and such that $f(i)=-2$. Any hint is welcomed.

Here is a hint: consider $f'(z)/z$. 
Since $f(z)$ has a zero of order two at $z=0$, the derivative $f'(z)$ is also holomorphic, and $f'(0)=0$. Thus, you may write $f'(z)$ as $z\cdot g(z)$, with $g(z)$ holomorphic. Then, the bound in the statement tells you that $|g(z)|$ is bounded.
