 Hölder factorization of stochastic exponentials

For a zero-starting continuous local martingale $X$, the stochastic exponential satisfiesThe Holder inequality and the supermartingale property of the first exponential bound its stopped $p$th moment by $(\mathbb{E}{e}^{C(p,q)X_T}{)}^{(q-1)/q}$. The subtraction in the coefficient is $\sqrt{pq}-1$, not a square root of $pq-1$.