How can the winding number change under a holomorphic map?

Question

This question comes from an old complex analysis qual. First denote $\mathbb{C}^{\times} = \mathbb{C} \backslash \{ 0 \}$, $u = \{ e^{it} : 0 \leq t < 2 \pi \}$, and let $f : \mathbb{C}^{\times} \to \mathbb{C}^{\times}$ be some holomorphic function. Then the winding number $$ \frac{1}{2 \pi i} \int_{f(u)} \frac{dz}{z} $$ can take any integer value depending on $f$ (by letting $f(z) = z^n$, for example). However, if we change the domain and codomain of $f$ to something like $\{ z \in \mathbb{C} : \frac{1}{4} < |z| < 4 \}$, then we can't use those $f$'s we used before. The winding number can still be $1$ of course by using the identity. I see that it can also be $0$ by letting $f(z) = 1$. We can also get the winding number to be $-1$ by setting $f(z) = z^{-1}$. Can we get the winding number to be $2$? I suspect not but am having trouble showing it. Thoughts?

Answer 1

This proof uses some basic facts about Extremal length quoted from the wikipedia article.

The extremal length $EL(\Gamma)$ of a collection of curves $\Gamma$ within some domain $D$ is defined to be

$$EL(\Gamma)=\sup_{\rho\in [0,\infty]^D}\frac{\left(\inf_{\gamma\in\Gamma}\int_\gamma\rho\,|dz|\right)^2}{\int_D\rho^2\,dx\,dy},$$

and the most important property of this quantity is that it is invariant under conformal maps, which is to say, if $\Gamma^*=\{f\circ\gamma:\gamma\in\Gamma\}$ for some bijective holomorphic function $f$, then $EL(\Gamma^*)=EL(\Gamma)$.

The wikipedia article has already gone to the trouble to calculate $$EL(\Gamma)=\frac{2\pi}{\log(r_2/r_1)}$$ for the collection $\Gamma$ of curves that wind once around the annulus of radii $r_1,r_2$, and $EL(\Gamma)=w/h$ for the collection of curves in a rectangle $w\times h$ that travel from one edge to the opposite edge (traveling a distance $w$).

Being an infimum over $\Gamma$, it is clear that $\Gamma\subseteq\Gamma'$ implies $EL(\Gamma)\ge EL(\Gamma')$. So now let us suppose that there is some holomorphic $f$ whose domain is the annulus $r_1,r_2$ and with codomain in this same annulus, and with a winding number $N>1$. We can "unwind" this function to $g=\log\circ f$, where $g$ analytically continues $\log$ so that the result is a bijective holomorphic function with codomain $(\log r_1,\log r_2)\times [0,2\pi N]$.

Every path $\gamma\in\Gamma_1$ is homotopic to $e^{it}:t\in[0,2\pi]$, and we know that the image of this path winds $N$ times around the annulus, so $g\circ\gamma\in\Gamma''$, the set of paths that got from $(\log r_1,\log r_2)\times\{0\}$ to $(\log r_1,\log r_2)\times\{2\pi N\}$, while staying within $(\log r_1,\log r_2)\times (-\infty,\infty)$, because of the codomain restriction on $f$.

Furthermore, for each curve $\gamma''\in\Gamma''$, some subset $\gamma'$ of it (with smaller path length) will be strictly contained in $(\log r_1,\log r_2)\times [0,2\pi N]$. Thus $\gamma'\in\Gamma'$, where $\Gamma'$ is the set of curves that cross from the bottom to the top of $(\log r_1,\log r_2)\times [0,2\pi N]$. so $EL(\Gamma'')\ge EL(\Gamma')$. This implies, from the above calculations:

$$EL(\Gamma_1)=EL(g\circ\Gamma)\ge EL(\Gamma'')\ge EL(\Gamma')\implies \frac{2\pi}{\log(r_2/r_1)}\ge \frac{2\pi N}{\log r_2-\log r_1},$$

which is a contradiction. (Also, given a holomorphic function $f$ with winding number $N<-1$, the composition with $z^{-1}$ satisfies the same conditions as above and so this is also impossible.)