Half-threshold for the exponential-martingale Hölder bound (${\inf }_{p,q>1}(pq-\sqrt{pq})/(q-1)=1/2$)

For fixed $q>1$, the coefficient in the Hölder factorization of stochastic exponentials exceeds $\sqrt{q}/(\sqrt{q}+1)>1/2$. Taking $q=1+\epsilon$ and $p=1+{\epsilon }^2$ approaches one-half. Thus a uniform stopped exponential moment at coefficient one-half bounds some $p$th moment of each strict scaling $\mathcal{E}(aM)$, $0<a<1$.