  In this paper, we generalize the completely monotone conjecture from Shannon
entropy to the R\'enyi entropy. We confirm this conjecture for the order of
derivative up to $3$, when the order of R\'enyi entropy is in certain regimes.
We also investigate concavity of R\'enyi entropy power and the completely
monotone conjecture for Tsallis entropy. We observe that the completely
monotone conjecture is true for Tsallis entropy of order $2$. Our proofs in
this paper are based on the techniques of integration-by-parts and
sum-of-squares.
