= One-point value region of a holomorphic disc map
{title2=$|\psi_{w_0}(w)|\le|\psi_{z_0}(z)|$}
For <holomorphic functions> $f:\mathbb D\to\mathbb D$ with $f(z_0)=w_0$, put $r=|(z-z_0)/(1-\overline z_0z)|$. Their possible values at $z$ form the closed disc with centre $(1-r^2)w_0/(1-r^2|w_0|^2)$ and radius $r(1-|w_0|^2)/(1-r^2|w_0|^2)$. Indeed $\psi_{w_0}\circ f=\psi_{z_0}h$ with <holomorphic> $|h|\le1$; all constant choices of $h$ in the closed <unit disc> are admissible. Completing the square in the displayed pseudohyperbolic inequality gives the centre and radius.
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