= Starting-value obstruction in wedge conjugate maximal inequalities
{title2=$P(0)=0\text{ does not imply }\operatorname{Im}P(i^\alpha)=0$}
Real and imaginary parts of an analytic Brownian image, after subtracting their starting values, have equal <quadratic variation>. The <Burkholder-Davis-Gundy inequalities> compare their centered maximal moments with constants depending only on $p$. A boundary-point normalization $P(0)=0$ does not remove the starting imaginary value. For $P(z)=iz$ on a wedge of opening $\alpha\pi<\pi/2$, $\sup|\operatorname{Im}z|\leq\tan(\alpha\pi)\sup|\operatorname{Re}z|$. If $\alpha p<1$ both relevant moments are finite, so an uncentered comparison requires a constant at least $\cot^p(\alpha\pi)$; no bound depending only on $p$ can hold over all such wedges. For fixed $\alpha>0$, <polynomials> with finite real-part maximal moment have degree $d<1/(\alpha p)$ by the wedge radial-maximum tail. They form a <finite-dimensional vector space>, and <equivalence of norms in finite dimensions> gives a valid constant depending on $p,\alpha$.
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