Moment ratios of centered log-concave random variables]{For exponents at least two, the centered exponential maximizes the moment ratios of centered log-concave random variables

For a random variable X with mean zero and a log-concave density, and for exponents 1≤ p≤ q, the ratio \|X\|q/\|X\|ₚ of its Lq and Lₚ norms is at most a constant that depends only on p and q. Eitan showed in 2024 that for $1<p<q$ the largest value of the ratio is attained within the one-parameter family Zₛ=s(E-1)-(1-s)(E'-1), 0≤ s≤1, where E and $E'$ are independent standard exponential random variables. He conjectured that for all positive even integers $p<q$ the largest ratio is attained by the centered exponential $E-1$ and its reflection $1-E$, and by no other law up to scaling. He confirmed that the maximizing s is $0$ or $1$ for even $p<q$ with $p=2$ or $q<100$. We prove that for all real p and q with 2≤ p<q and q≥4 the largest ratio is that of $E-1$, and that only positive multiples of $E-1$ and of $1-E$ attain it. Positive even integers $p<q$ satisfy p≥2 and q≥4, so the case of even $p<q$ is Eitan's conjecture. For each exponent $n>1$ and each s∈(0,1/2), the proof writes the derivative of log\|Zₛ\|ₙ with respect to s as [ₛ(|Zₛ|)|Zₛ|ⁿ]/|Zₛ|ⁿ for an explicit function ₛ on (0,∞), and shows that ₛ increases and then decreases. Hence the derivative is a strictly quasi-concave function of n, by an argument about the sign changes of ₛ minus a constant. The derivative is also smaller at $n=4$ than at $n=2$, so, by quasi-concavity, it is smaller at $n=q$ than at $n=p$ when q≥4, which makes \|Zₛ\|q/\|Zₛ\|ₚ strictly decreasing for 0≤ s≤1/2. One further inequality in one real variable, the only step of the proof that a computer certifies, removes the condition q≥4.