Find all holomorphic functions with the following property

Question

Let $D$ be the unit disc. Find all holomorphic functions $f:D\to D$ such that $f(\frac14)=\frac14$, and $f'(\frac14)=\frac7{15}$.

I guess that we should use Schwarz lemma. And I guess that the only solution is the linear one.

Answer 1

Hint: Let $g(z) = \frac{1/4 -z}{1-(1/4)z}.$ I assume you know this map is a holomorphic bijection of $D$ onto $D.$ Note $g(0)= 1/4$ and $g(1/4)=0.$ Apply the Schwarz Lemma to $g\circ f \circ g.$