= Solution
{c}
A bounded <holomorphic function> of planar Brownian motion is a local martingale up to exit, because its real and imaginary parts are harmonic and <Itô's formula> has zero drift. Stop before the given $S<T_D$. Localization inside compact subdisks gives the local martingale $f(Z_{t\wedge S}^z)$, and $|f|<1$ makes it bounded. The <bounded local martingale criterion> therefore gives a true martingale.
Exit from the unit disk is finite almost surely, so $S<T_D$ is finite too. Taking $t\to\infty$ and using the <dominated convergence theorem> yields
$$
\boxed{\mathbb E[f(Z_S^z)]=f(z).}
$$
The same proof works for any bounded <holomorphic function>, after rescaling if needed; it does not require the stopping time itself to be bounded.
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