New bounds for the ratio of power means

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