Half-plane capacity versus harmonic hull capacity ($\mathrm{hcap}(K)\le \mathrm{rad}(K)\mathrm{cap}(K)/\pi$)

The Brownian representation of half-plane capacity is $\mathrm{hcap}(K)={\lim }_{y\to \infty }y{\mathbb{E}}_{iy}\mathrm{Im}{B}_T$. The harmonic capacity from infinity in the upper half-plane uses $\mathrm{cap}(K)={\lim }_{y\to \infty }\pi y{\mathbb{P}}_{iy}(B_T\in K)$. Since $0\le \mathrm{Im}{B}_T\le \mathrm{rad}(K)1_{\{B_T\in K\}}$, the displayed bound follows. The normalization factor $\pi$ cannot be omitted. Independently, enclosure in a half-disc and monotonicity of half-plane capacity give the sharp bound $\mathrm{hcap}(K)\le \mathrm{rad}(K{)}^2$.