= Concentric normalization of an eccentric annulus
{title2=$T_a(z)=(z-a)/(1-az)$}
For an inner circle of real center $c$ and radius $d$ lying strictly inside the unit circle, choose the real root $|a|<1$ of $ca^2-(1+c^2-d^2)a+c=0$. When $c=0$, take $a=0$. The <automorphism of the unit disk> $T_a$ sends the inner circle to a concentric circle. Indeed it sends the two real endpoints $c-d,c+d$ to opposite real numbers; preservation of circles and reflection symmetry then give an image centered at zero. Its radius is $r=T_a(c+d)\in(0,1)$.
The <conformal invariance of harmonic measure> and <harmonic measure of an annulus> give outer-boundary measure $\log(|T_a(z)|/r)/\log(1/r)$. For $c=d=2/5$, the quadratic gives $a=1/2$ and $r=1/2$, an explicit eccentric-annulus example.
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