I want to find a conformal mapping of the unit disk $\mathbb{D}$ onto itslef that takes 1/2 to 1/3. Here is my attempt:
We know that $f(z)=\frac{z-a}{\bar{a}z-1}$ with $|a|<1$ maps $\mathbb{D}$ onto itself. So, if we let $f(z)=\frac{1}{3}$ and $z=\frac{1}{2}$, then algebraically we can solve to get $a=\frac{5}{7}$. Therefore the map $f(z)=\frac{z-\frac{5}{7}}{\frac{5z}{7}-1}$ will do the trick.
Would this work? Thank you!
Your computation of a possible value of $a$ is correct. But your error is to have written that the conjugate of $5/7$ is $-5/7$ instead of $5/7$ plainly.
Therefore the function is $$f(z)=\dfrac{z-\tfrac57}{\tfrac57 z -1}$$
Here is a way to have an idea of how function $f$ operates (on the left the disk with concentric circles and radial lines orthogonal to them ; on the right, their images by function $f$. Please note that orthogonality is preserved. Point $1/2$ and its image $1/3$ have been represented by a little star.