On the Completely Monotone Conjecture for R\'enyi Entropy

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.