Existence of a holomorphic function with value of the function and the value of the differentiated function are given at two distinct points.

Question

Let $\mathbb D $ = {z $\in$ $\mathbb C$ : $\vert z|$ $\lt 1$} .Then does there exist a holomorphic function $f$ : $\mathbb D $ $\rightarrow$ $ \mathbb D$ with $f(\frac{3}{4})$ = $\frac{3}{4}$ and $ g(\frac{2}{3})$ =$ \frac{3}{4}$ where g(z) is the differentiation of f(z) (actually I don't know how to write differentiation using dashe ). If both the points are same then it's easy to solve.But how about this ? Hope to get help. Thank you in advance.

Answer 1

$f(z)=\frac{3}{4}z+\frac{3}{16}$ works !

I used the "Ansatz": $f(z)=az+b$.