Half-threshold for the exponential-martingale Hölder bound
= Half-threshold for the exponential-martingale Hölder bound
{title2=$\inf_{p,q>1}(pq-\sqrt{pq})/(q-1)=1/2$}
For fixed $q>1$, the coefficient in the <Hölder factorization of stochastic exponentials> exceeds $\sqrt q/(\sqrt q+1)>1/2$. Taking $q=1+\varepsilon$ and $p=1+\varepsilon^2$ approaches one-half. Thus a uniform stopped exponential moment at coefficient one-half bounds some $p$th moment of each strict scaling $\mathcal E(aM)$, $0<a<1$.