a property of Analytic functions Let $D$ be the open unit disk. How can I find all analytic functions $f: D\longrightarrow D$ such that $f(\frac 14) = \frac 14$ and $f'(\frac 14) = \frac 7{15}$ ?
 A: Let $$\phi(z) = \frac{a-z}{1-\bar a z}, $$ where $a= \frac14$. (This is a standard trick. Note that $\phi : D \to D$ with $\phi(a) = 0$ and $\phi(0) = a$.) Put $g(z) = \phi(f(\phi(z)))$. Then $g : D \to D$ and
$$g(0) = \phi(f(\phi(0)) = \phi(f(a)) = \phi(a) = 0,$$
Also
$$
g'(z) = \phi'(f(\phi(z)) f'(\phi(z)) \phi'(z),
$$
so
$$
g'(0) = \phi'(a) f'(a) \phi'(0).
$$
Computing
$$ \phi'(z) = \frac{1-|a|^2}{(1-\bar a z)^2},$$
we see that
$$
g'(0) = \frac{1}{(1-|a|^2)} \cdot f'(a) \cdot (1-|a|^2) = f'(a).
$$
It remains to find all analytic functions $g: D \to D$ with $g(0) = 0$ and $g'(0) = \frac{7}{15}$. Put $h(z) = g(z)/z$. Then $h : D \to D$ and $h(0) = 7/15$. 
Now we have many choices. If $h(z) = 7/15$, untangling everything we end up with
$$ f(z) = \frac{97z+32}{233-32z},$$
and you can check that this satsifies the assumptions.
I'll leave it up to you to find a description of all possible choices of $h$. You can argue in a similar way as above, and use Schwarz' lemma to get some restrictions on $h$.
