  The motivation for this paper comes from the following question on comparison
of norms of conformal martingales $X$, $Y$ in $\R^d$, $d\geq 2$. Suppose that
$Y$ is differentially subordinate to $X$. For $0<p<\infty$, what is the optimal
value of the constant $C_{p,d}$ in the inequality $$ Y_p\leq C_{p,d}X_p ?$$ We
answer this question by considering a more general related problem for
nonnegative submartingales. This enables us to study extension of the above
inequality to the case when $d>1$ is not an integer, which has further
interesting applications to stopped Bessel processes and to the behavior of
smooth functions on Euclidean domains. The inequality for conformal
martingales, which has its roots on the study of the $L^p$ norms of the
Beurling-Ahlfors singular integral operator \cite{BW}, extends a recent result
of Borichev, Janakiraman and Volberg \cite{BJV2}.
