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 $1\le p\le q$, the ratio $\|X\|_q/\|X\|_p$ of its $L_q$ and $L_p$ 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 $Z_s=s(E-1)-(1-s)(E'-1)$, $0\le s\le1$, 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 $2\le p<q$ and $q\ge4$ 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 $p\ge2$ and $q\ge4$, so the case of even $p<q$ is Eitan's conjecture. For each exponent $n>1$ and each $s\in(0,1/2)$, the proof writes the derivative of $\log\|Z_s\|_n$ with respect to $s$ as $\E[\gbar_s(|Z_s|)|Z_s|^n]/\E|Z_s|^n$ for an explicit function $\gbar_s$ on $(0,\infty)$, and shows that $\gbar_s$ increases and then decreases. Hence the derivative is a strictly quasi-concave function of $n$, by an argument about the sign changes of $\gbar_s$ 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 $q\ge4$, which makes $\|Z_s\|_q/\|Z_s\|_p$ strictly decreasing for $0\le s\le1/2$. One further inequality in one real variable, the only step of the proof that a computer certifies, removes the condition $q\ge4$.