A reverse entropy power inequality for i.i.d. log-concave random variables

We show that $h_{\mathrm{\infty }}(X+Y)\le h_{\mathrm{\infty }}(Z+W)$, where $X,Y$ are independent log-concave random variables, and $Z,W$ are exponential random variables having the same respective $\mathrm{\infty }$-Rényi entropies. Analogs for integer-valued monotone log-concave random variables are also obtained. Our main tools are decreasing rearrangement, majorization, and the change of measure.