Suppose $f$ is analytic in $|z|<2$ and $|f(z)|\le 2$ there, and $f(1)=0$. Find the best possible bound for $f(1/4)$.
I guess 'the best possible bound' means that the bound is as close as possible to the supremum of all possible values of $f(1/4)$. However, I can't figure out the best possible bound by the given data. It is clear that the bound is at most 2, but how can I find the smaller bound for this value?
Let $D$ be the unit disk. The Schwartz lemma tells you that any holomorphic $f : D \to D$ such that $f(0) = 0$ obeys $|f(z)| \leq |z|$ for all $z \in D$. This inequality is saturated by the function $f(z) = z$.
You can easily scale this result up to a disk of radius 2.
A difficulty that you face is that your $f$ sends $1 \mapsto 0$ rather than $0 \mapsto 0$. To deal with this, you could compose your $f$ with a conformal map sending the disk to itself in such a way that $0 \mapsto 1$.
Note that a standard conformal mapping from $D \to D$ sending $\alpha \mapsto 0$ is $$ z \mapsto \frac {z - \alpha} { 1- \bar \alpha z},$$ though again you need to scale this up to disks of radius 2.
I hope this is enough of a hint.