= Conformal radius under a chordal Loewner flow
{title2=$\Upsilon_t=\operatorname{Im}g_t(z)/|g_t'(z)|$}
= Loewner conformal radius
{c}
{synonym}
Before the <Loewner swallowing time> of $z$, set $Z_t=g_t(z)-U_t=X_t+iY_t$ and $\Upsilon_t=Y_t/|g_t'(z)|$. Since the <conformal radius> of the <complex upper half-plane> at $Z_t$ is $2Y_t$, the transformation rule gives $2\Upsilon_t=\operatorname{crad}_{D_t}(z)$. The <Koebe quarter theorem> and the interior-disc comparison give $\operatorname{dist}(z,\partial D_t)/2\leq\Upsilon_t\leq2\operatorname{dist}(z,\partial D_t)$. Differentiating the <Chordal Loewner equation> gives $\boxed{\partial_t\log\Upsilon_t=-4Y_t^2/|Z_t|^4}$. Thus this quantity decreases as the remaining <domain> shrinks.
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