###### Definition.

Let $G\subset{\mathbb{C}}$ be a simply connected region that is not the whole plane and let $a\in G$ be any point. The Riemann mapping theorem  tells us that there exists a unique one-to-one and onto holomorphic map $f\colon{\mathbb{D}}\to G$ (where ${\mathbb{D}}$ is the unit disc) such that $f(0)=a$ and $f^{\prime}(0)>0$. Then define the $r(G,a)=f^{\prime}(0)$.

###### Example.

For example, take $G=B(0,\delta)$ (the open ball of radius $\delta$ around 0) for some $\delta>0$, then $r(G,0)=\delta$ because we have a map $f(z)=\delta\cdot z$ as our unique map. And thus this definition coincides with our definition of radius for this special case.

###### Example.

For another example we look at how the conformal radius is affected by the choice of the point $a$. So suppose that we take $G$ to be the unit disc (${\mathbb{D}}$) itself and we take some point $a\in{\mathbb{D}}$. The unique map that takes 0 to $a$ is the map $f(z)=\frac{z+a}{1+\bar{a}z}$ (where $\bar{a}$ is the complex conjugate   of $a$) and by the quotient rule we get that $f^{\prime}(z)=\frac{1-\lvert a\rvert^{2}}{(1+\bar{a}z)^{2}}$. And so $r({\mathbb{D}},a)=f^{\prime}(0)=1-\lvert a\rvert^{2}$, so the conformal radius of the unit disc goes to 0 as we move the point $a$ towards the boundary of the disc, and it is largest (equal to 1) when $a=0$.

From the first example we can now see another way of characterizing the conformal radius. Take the inverse map (inverses of holomorphic one-to-one functions are also always holomorphic) and call it $\varphi\colon G\to{\mathbb{D}}$ (the map such that $\varphi(f(z))=z$). We take the derivative (see the entry on univalent functions  (http://planetmath.org/UnivalentFunction)) we get $\varphi^{\prime}(f(0))=\frac{1}{f^{\prime}(0)}$, that is $\varphi^{\prime}(a)=\frac{1}{r}$ (where we call $r=r(G,a)$ for brevity now). If we multiply the map by the conformal radius we get a map $\gamma\colon G\to B(0,r)$ such that $\gamma(z)=r\cdot\varphi(z)$ and $\gamma^{\prime}(a)=1$. By uniqueness of the map arising from the Riemann mapping theorem we can see that $\gamma$ is also unique. Thus we could define the conformal radius as follows.

###### Definition.

Let $G\subset{\mathbb{C}}$ be a region and let $a\in G$ be any point. By application of Riemann mapping theorem there exists a unique map $\gamma\colon G\to B(0,r)$ for some $r>0$, such that $\gamma(a)=0$ and $\gamma^{\prime}(a)=1$. The conformal radius is then defined as $r(G,a)=r$.

This definition gives more of an intuitive understanding of why we’d call this the conformal radius of $G$. We look at the unique map with $\gamma^{\prime}(a)=1$, that is, the map that doesn’t “stretch” the set. So the radius of $G$ with respect to $a$ is really the radius of the unique ball around zero to which $G$ is conformally equivalent without any “stretching” needed.

## References

• 1 S. Rohde, M. Zinsmeister. , Journal d’Analyse (to appear). Available at http://www.math.washington.edu/ rohde/papers/rozi.pshttp://www.math.washington.edu/ rohde/papers/rozi.ps
Title conformal radius ConformalRadius 2013-03-22 14:18:33 2013-03-22 14:18:33 jirka (4157) jirka (4157) 9 jirka (4157) Definition msc 30C55 RiemannMappingTheorem