Terminal scaling inequality for stochastic exponentials ($\mathbb{E}\mathcal{E}(rX{)}_{\infty }\ge (\mathbb{E}\mathcal{E}(X{)}_{\infty }{)}^{r^2}(\mathbb{E}{e}^{r{X}_{\infty }/2}{)}^{-2(r-1)}$)

For a convergent continuous local martingale and $r>1$, factor $\mathcal{E}(X)$ into $\mathcal{E}(rX{)}^{1/r^2}$ and an ordinary exponential. Apply the Holder inequality and then the concave Jensen inequality to the fractional power $2/(r+1)$. The displayed lower bound is informative when the terminal exponential moment is finite.