= Solution
{c}
For $0<\alpha<2$, write $z=re^{i\theta}$ with $0<\theta<\pi$. The chosen power map sends this to $r^\alpha e^{i\alpha\theta}$, so
$$
\boxed{D=\{re^{i\theta}:r>0,\ 0<\theta<\alpha\pi\}.}
$$
At $\alpha=2$ this is the plane slit along the nonnegative real axis. For every $0<\alpha\leq2$, the map is conformal on the upper half-plane.
To check finiteness of the physical exit time, start a <Brownian motion> $B$ at $i$ and let $T$ be its real-axis exit. The <time change of a continuous process> in <conformal invariance of planar Brownian motion> is
$$
A_T=\int_0^T\alpha^2|B_s|^{2\alpha-2}ds.
$$
We know $T<\infty$ almost surely. The <Cauchy exit law from a Brownian half-plane> assigns <probability> zero to $B_T=0$. On such a path, $B_s\neq0$ throughout the compact interval $[0,T]$, because it is in the open upper half-plane before exit and has nonzero endpoint. Its modulus therefore has a positive minimum and finite maximum. The displayed integrand is bounded on that interval, proving $A_T<\infty$. The image <Brownian motion> exits $D$ at this finite changed time.
The endpoint $\alpha=0$ printed in the source is degenerate: $z^0=1$, so its image is the singleton $\{1\}$, not an open domain, and the map is not conformal. If exit from that set is defined for <Brownian motion> started at one, its exit infimum is zero. It cannot be included in a nondegenerate conformal-domain argument.
Here is the <martingale> argument for maximal comparisons, including the starting-value issue. For $P=f+ig$, the <Cauchy-Riemann equations> and <Itô's formula> show that
$$
F_t=f(Z_{t\wedge\tau})-f(i^\alpha),\qquad
G_t=g(Z_{t\wedge\tau})-g(i^\alpha)
$$
are orthogonal continuous <local martingales> with the same <quadratic variation>. Localize before large disks and before exit. The <Burkholder-Davis-Gundy inequalities> then give
$$
\mathbb E(G^*)^p\leq C_p\mathbb E(F^*)^p,\qquad p>1,
$$
with a constant depending only on $p$; <Fatou's lemma> removes the localization. These are centered <martingales>. For the uncentered processes this implies
$$
\mathbb E\sup|g(Z_t)|^p
\leq C_p'\bigl(\mathbb E\sup|f(Z_t)|^p+|g(i^\alpha)|^p\bigr),
$$
since $|f(i^\alpha)|\leq\sup|f(Z_t)|$.
The source normalizes $P(0)=0$ at a boundary point, rather than the value of $g$ at the starting point. Consequently its displayed constant cannot be uniform in $\alpha$ with only $p$ dependence. To see this, take $P(z)=iz$ and $0<\alpha<\min(1/2,1/p)$. Then $f=-\operatorname{Im}z$ and $g=\operatorname{Re}z$. Inside this narrow wedge,
$$
\sup|f(Z_t)|\leq\tan(\alpha\pi)\sup|g(Z_t)|.
$$
Both $p$th moments are finite and the right one is positive; finiteness follows from the wedge radial-maximum tail of order $R^{-1/\alpha}$, justified below. The source inequality would force $C_p\geq\cot^p(\alpha\pi)$, unbounded as $\alpha\downarrow0$. At $\alpha=0$, the immediate-exit interpretation gives the direct contradiction $1\leq C_p\,0$ for the same <polynomial>.
For a fixed nondegenerate wedge, an uncentered inequality does hold with a constant allowed to depend on $\alpha$ as well as $p$. One rigorous outline for the <polynomial> class in this question is as follows. Separation of variables in a sector of radius $R$ gives, for exit through its circular arc, the expansion
$$
\mathbb P_{re^{i\theta}}(\text{hit radius }R\text{ before the sides})
=\frac4\pi\sum_{\substack{n\geq1\\n\ {\rm odd}}}
\frac1n(r/R)^{n/\alpha}\sin(n\theta/\alpha).
$$
Thus at the specified starting point the radial-maximum tail is comparable to $R^{-1/\alpha}$. Hitting any fixed interior subarc has that same power, with positive leading coefficient. If a nonzero <polynomial> has degree $d$, its <real part> has magnitude at least $cR^d$ on some fixed interior subarc for sufficiently large $R$. Therefore its maximal real-part $p$th moment is infinite whenever $dp\geq1/\alpha$. In that case the desired extended-real inequality is automatic.
The remaining <polynomials> have $d<1/(\alpha p)$, so belong to a <finite-dimensional vector space> space. On that space
$$
P\longmapsto\left(\mathbb E\sup|f(Z_t)|^p\right)^{1/p}
$$
is a genuine <norm>: if it is zero, <Brownian motion>'s positive <probability> of reaching any interior neighborhood forces $\operatorname{Re}P=0$ throughout $D$, so analyticity makes $P$ a purely imaginary constant, and $P(0)=0$ makes it zero. The corresponding imaginary-part functional is bounded on this <finite-dimensional vector space> normed space; its finiteness follows from $dp<1/\alpha$ and the radial tail bound. <equivalence of norms in finite dimensions> gives
$$
\boxed{\mathbb E\sup|g(Z_t)|^p
\leq C_{p,\alpha}\mathbb E\sup|f(Z_t)|^p
\quad(0<\alpha\leq2).}
$$
This establishes the fixed-domain interpretation and identifies exactly why the universal centered <martingale> inequality cannot silently be applied to the printed uncentered boundary normalization. It is the distinction in <starting-value obstruction in wedge conjugate maximal inequalities>.
\Image[/past-exam-of-the-stochastic-differential-equations-and-brownian-motion-in-complex-analysis-course-of-the-university-of-oxford/2008-power-wedges.png]
{title=The power map changes the wedge angle and degenerates to a constant at zero exponent}
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