Rényi Entropy and Variance Comparison for Symmetric Log-Concave Random Variables

We show that for any
$\alpha >0$
the Rényi entropy of order
$\alpha $
is minimized, among all symmetric log-concave random variables with fixed variance, either for a uniform distribution or for a two sided exponential distribution. The first case occurs for
$\alpha \in (0,\alpha ^{\ast}]$
and the second case for
$\alpha \in [\alpha ^{\ast},\infty$
), where
$\alpha ^{\ast}$
satisfies the equation
$2 \log \alpha ^{\ast}= (\alpha ^{\ast}-1) \log 6$
, that is
$\alpha ^{\ast} \approx 1.241$
. We deduce that the one-sided exponential distribution minimizes Rényi entropy of order
$\alpha \geq 2$
among all log-concave random variables with fixed variance.