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A chordal Loewner chain is an increasing family of compact H-hulls, usually parameterized by half-plane capacity, whose mapping-out functions evolve according to the Chordal Loewner equation.

A compact H-hull is a bounded relatively closed set $A\subset \mathbb{H}$ for which $\mathbb{H}\setminus A$ is a simply connected domain. The Riemann mapping theorem and hydrodynamic normalization at infinity give a unique mapping-out function of a compact H-hull $g_A:\mathbb{H}\setminus A\to \mathbb{H}$ withThe coefficient $a\ge 0$ is the half-plane capacity $\mathrm{hcap}(A)$.

For $\lambda >0$ and $x\in \mathbb{R}$, uniqueness of the normalized map givesSubstituting the expansion of $g_A$ yieldsTherefore the scaling and translation of half-plane capacity is

Let $d=\mathrm{diam}(A)$. Unless $A$ is empty, its closure meets the real axis; otherwise a loop in $\mathbb{H}\setminus A$ surrounding $A$ could not contract, contrary to $\mathbb{H}\setminus A$ being a simply connected domain. Choose $x\in \overline{A}\cap \mathbb{R}$. Then $A\subseteq \{z\in \mathbb{H}:|z-x|\le d\}$.

The unit half-disc is a compact H-hull with mapping-out function $z+z^{-1}$, so its half-plane capacity is one. The monotonicity of half-plane capacity and its scaling rule now giveThus the assertion holds with the universal constant $c=1$ under this normalization.