= Solution
Write $r_t=\sqrt{2t}$. The hulls $A_t=r_t(\mathbb H\cap\mathbb D)$ are nested, so property (i) holds. Scaling the given map gives
$$
g_t(z)=z+\frac{r_t^2}{z}=z+\frac{2t}{z},
$$
so $\operatorname{hcap}(A_t)=2t$ and property (ii) holds.
Property (iii) fails. For $s>0$, the image under $g_s$ of the outer semicircle of $A_t\setminus A_s$ is
$$
g_s(r_te^{i\theta})
=r_te^{i\theta}+\frac{r_s^2}{r_te^{i\theta}}.
$$
As $t\downarrow s$, this converges to $2r_s\cos\theta$, which fills the real interval $[-2r_s,2r_s]$. Hence the diameter of the mapped new increment tends to $4r_s$, rather than zero. Therefore \b[(i) and (ii) hold, while (iii) does not].
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