As part of a recent homework set in my complex analysis course, I was tasked with a problem that required a slight generalization on part of Schwarz’s Lemma:

Lemma (Schwarz):Let $f$ be analytic on the unit disk with $|f(z)| \leq 1$ for all $z$ on the disk and $f(0) = 0$. Then $|f(z)| < |z|$ and $f’(0)\leq 1$.

If either $|f(z)|=|z|$ for some $z\neq0$ or if $|f’(0)|=1$, then $f$ is a rotation, i.e., $f(z)=az$ for some complex constant $a$ with $|a|=1$.

The homework assignment asked us (within the context of a larger problem) to consider the case when $f(\zeta) = 0$ for some $\zeta \neq 0$ on the interior of the unit disk. The secret to this problem was to find some analytic function $\varphi$ that maps the unit disk to itself, but *swaps* $0$ and $\zeta$. Then, we may consider $\varphi^{-1}\circ f\circ \varphi$ and apply Schwarz’s Lemma.

## Properties of the transformation

The appropriate map, which is a particular Möbius transformation, is given by the following:

$$\varphi_\zeta(z) = \frac{\zeta - z}{1-\overline{\zeta}z}$$

Now, if $|z| = 1$, then $|\varphi_\zeta(z)| = |\overline{z} \varphi_\zeta(z)| = \left|\frac{\overline{z}\zeta-1}{1-\overline{\zeta}z}\right| = 1$. Therefore, this map takes the boundary of the unit disk to itself.

Further, this $\varphi_\zeta$ is analytic within the unit disk, as its only singularity occurs when $|z| > 1$ (since this occurs when $z = \frac{1}{\overline{\zeta}}$ and $\left|\overline{\zeta}\right| < 1$). And, finally, since $\varphi_\zeta$ is non-constant, the maximum modulus principle tells us that $|\varphi_\zeta(z)| < 1$ when $|z| < 1$.

Therefore, $\varphi_\zeta$ maps the unit disk onto itself, where $\varphi_\zeta(\zeta) = 0$ and $\varphi_\zeta(0) = \zeta$.

Another useful feature of this map is that it is an involution. That is, $\varphi_\zeta^{-1} = \varphi_\zeta$. An application of Schwarz’s Lemma shows this immediately: since $\varphi\circ\varphi$ fixes *two* points in the unit disk (one of them zero) and satisfies the modulus bound, we can conclude that $\varphi\circ\varphi$ is the identity. Therefore, $\varphi$ is its own inverse.

## Impact of this map on the unit disk

I was curious what this mapping does to the values on the unit disk. We’ve clearly swapped $\zeta$ and $0$, but the map must maintain analyticity on the unit disk, so it must do more than just that. I wanted to know how this distortion affects the rest of the values on the disk. So, I wrote a quick Python program to generate a couple of plots:

1 | import numpy as np |

Below, we’ve plotted the magnitude and argument (angle) of $z$ and $\varphi_\zeta(z)$ side-by-side. We can now see that, in terms of magnitude, it’s just as if the map “shifted” over the origin, squeezing and pulling the surrounding values to maintain analyticity. However, it also *twisted and reflected* the values of $z$ on each circle around the origin. (This can be seen through the curve of the $\mathrm{Arg}(\varphi_\zeta(z))$ plot.)

## Conclusion

This doesn’t really serve much purpose in and of itself, but it helped build my intuition of what is happening when I apply the function $\varphi$ and developed my abilities in `numpy`

and `matplotlib`

usage. The Schwarz Lemma is an interesting topic in Complex Analysis, and I based some of my initial work on a 2010 paper by Dr. Harold P. Boas, entitled *Julius and Julia: Mastering the art of the Schwarz lemma*. Of particular note is “Section 3: Change of Base Point,” where he develops and discusses the map $\varphi$.