Local Diffeomorphism from $D_R$ to ${C}$

Question

I am trying to find a function that maps a disk of arbitrary radius $R$ to the complex plane. I have found that the function $$f(z)= \frac{z}{R}\frac{1}{1-|z/R|}$$ appears to do this precisely, however the magnitude on the bottom is causing me some difficulty in the problem I am attempting to solve. Ideally the $f(z)$ should be a diffeomorphism in the real sense! I was wondering if such a function $f(z)$ exists, but instead of considering $|z/R|$ on the denominator, we instead have $|z/R|^2$ such that the square root vanishes? Wikipeidia proposes that the function $$f(z)=\frac{z}{1-|z|^2}$$ is a bijective, real analytic function of the unit disk to the complex plane with analytic inverse. However I am not convinced with this since the inverse of this function is $$f^{-1}(z)=\frac{-1\pm\sqrt{4z^2+1}}{2z}$$ which is multivalued for some values of $z$.