New bounds for the ratio of power means

We show that for real numbers $p,\,q$ with $q<p$, and the related power means $\mathscr{P}_p$, $\mathscr{P}_q$, the inequality $$\frac{\mathscr{P}_p(x)}{\mathscr{P}_q(x)} \le \exp \bigg( \frac{p-q}8 \cdot \bigg(\ln\bigg(\frac{\max x}{\min x}\bigg)\bigg)^2 \:\bigg)$$ holds for every vector $x$ of positive reals. Moreover we prove that, for all such pairs $(p,q)$, the constant $\tfrac{p-q}8$ is sharp.