Comparing moments of real log-concave random variables

We show that for every mean zero log-concave real random variable $X$ one has $\mathrm{\parallel }X{\mathrm{\parallel }}_p\le \frac{p}{q}\mathrm{\parallel }X{\mathrm{\parallel }}_q$ for $p\ge q\ge 1$, going beyond the well-known case of symmetric random variables. We also prove that in the class of arbitrary log-concave real random variables for $p>q>0$ the quantity $\mathrm{\parallel }X{\mathrm{\parallel }}_p\mathrm{/}\mathrm{\parallel }X{\mathrm{\parallel }}_q$ is maximized for some shifted exponential distribution. Building upon this we derive the bound $\mathrm{\parallel }X{\mathrm{\parallel }}_p\le C_0\frac{p}{q}\mathrm{\parallel }X{\mathrm{\parallel }}_q$ for arbitrary log-concave $X$, with best possible absolute constant $C_0=e^{W(1\mathrm{/}e)}\approx 1.3211$ in front of $\frac{p}{q}$, where $W$ stands for the Lambert function.