Characterization of one-to-one conformal mapping from unit disc onto a square

complex-analysis conformal-geometry

Solution 1:

Hint: $g(z)=f^{-1}(if(z))$ is a conformal 1-1 map from $U$ onto itself fixing the origin. Try playing a little with this or something similar.

Related

Is a smooth immersion $c: [a,b] \to M$ injective if its image is a 1-manifold with non-empty boundary?
How to prove binomial coefficient $ {2^n \choose k} $ is even number?
Is it true that $|e^z|\le e^{|z|}$ for all $z \in \mathbb C$?
Why isn't Axiom of Choice a trivial result? Is it needed to prove existence of this recursive function?
Simple extension of $\mathbb{Q} (\sqrt[4]{2},i)$
Find the area of ​the $AFRC$ quadrilateral below.
Is the space of probability measures on a compact set is compact w.r.t Wasserstein metric?
Does the Jordan-Brouwer Separation Theorem require smoothness and orientability of the hypersurface?
How do negative powers and fractional powers make sense? [duplicate]
Proof of $\text{rank}(A+B)\leq \text{rank}(A)+\text{rank}(B)$ by another way.
Poisson process has probability with other distribution
Proving $End_A(V) $ is a local ring whose Jacobson -radical is a nil-radical ideal