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.

翻译：本文研究了沿热流的Rényi熵完全单调猜想。当Rényi熵的阶数处于特定区间时，我们证实了该猜想对导数阶数不超过$4$的情况。我们还研究了Rényi熵幂的凹性及Tsallis熵的完全单调性。我们恢复并略微扩展了Hung关于Tsallis熵四阶导数的结果，并观察到阶数为$2$的Tsallis熵具有完全单调性，这等价于热半群相关的噪声稳定性具有完全单调性。基于这一观察，我们猜想所有阶数$\alpha\in(1,2)$的Tsallis熵均满足完全单调性。本文的证明主要基于分部积分法、平方和分解与曲线拟合技术。