Comparing moments of real log-concave random variables

We show that for every mean zero log-concave real random variable $X$ one has $\|X\|_p \leq \frac{p}{q} \|X\|_q$ for $p \geq q \geq 1$, 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 $\|X\|_p / \|X\|_q$ is maximized for some shifted exponential distribution. Building upon this we derive the bound $\|X\|_p \leq C_0 \frac{p}{q} \|X\|_q$ for arbitrary log-concave $X$, with best possible absolute constant $C_0=e^{W(1/e)} \approx 1.3211$ in front of $\frac{p}{q}$, where $W$ stands for the Lambert function.