On the Completely Monotone Conjecture for Rényi Entropy

In this paper, we investigate the completely monotone conjecture along the heat flow for the R\'enyi entropy. We confirm this conjecture for the order of derivative up to $4$, when the order of R\'enyi entropy is in certain regimes. We also investigate concavity of R\'enyi entropy power and the complete monotonicity of Tsallis entropy. We recover and slightly extend Hung's result on the fourth-order derivative of the Tsallis entropy, and observe that the complete monotonicity holds for Tsallis entropy of order $2$, which is equivalent to that the noise stability with respect to the heat semigroup is completely monotone. Based on this observation, we conjecture that the complete monotonicity holds for Tsallis entropy of all orders $\alpha\in(1,2)$. Our proofs in this paper are based on the techniques of integration-by-parts, sum-of-squares, and curve-fitting.