In this paper, we investigate the complete monotonicity of R\'enyi entropy along the heat flow. We confirm this property 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Rényi熵沿热流的完全单调性。我们证实了当Rényi熵的阶数处于特定区间时，其导数阶数最高至$4$的完全单调性成立。我们还探讨了Rényi熵幂的凹性及Tsallis熵的完全单调性。我们重现并略微扩展了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，我们推测所有阶数$\alpha\in(1,2)$的Tsallis熵均满足完全单调性。本文的证明基于分部积分、平方和与曲线拟合等技术。