= Half-plane capacity versus harmonic hull capacity
{title2=$\operatorname{hcap}(K)\leq\operatorname{rad}(K)\operatorname{cap}(K)/\pi$}
The <Brownian representation of half-plane capacity> is $\operatorname{hcap}(K)=\lim_{y\to\infty}y\mathbb E_{iy}\operatorname{Im}B_T$. The <harmonic capacity from infinity in the upper half-plane> uses $\operatorname{cap}(K)=\lim_{y\to\infty}\pi y\mathbb P_{iy}(B_T\in K)$. Since $0\leq\operatorname{Im}B_T\leq\operatorname{rad}(K)\mathbf1_{\{B_T\in K\}}$, 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 $\operatorname{hcap}(K)\leq\operatorname{rad}(K)^2$.
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