The strong independent segment lemma …

In this post, we’re going to talk about a minor result in complex analysis, but it was a clue for a main result in this paper. 

Riemann mapping theorem

Riemann mapping theorem states that any two simply connected domains can communicate via a univalent function $f$. In other words, if $U$ and $V$ are two simply connected domains then there exists a one-to-one analytic map $f :U \rightarrow V$. 

Of course, such a theorem is not easy to prove and has a long history. If you are interested, I can advise you to have a look at this article of J. L Walsh.

The lemma

Let $U$ be a simply connected domain, symmetric with respect to the real axis. Let $L$ be the intersection between $U$ and the real axis.  Then any univalent map $f:\mathbb{D} \rightarrow U$ will map a certain diameter of $\mathbb{D}$ to $L$. In other words, there will be always a diameter that is not distorted by $f$. That is why, I’ve decided to call the result the strong independent segment lemma. 

If $f$ is a univalent function from $\mathbb{D}$ onto a symmetric simply connected domain $U$, then a certain diameter will not be distorted.