We utilize a discrete version of the notion of degree of freedom to prove a sharp min-entropy-variance inequality for integer valued log-concave random variables. More specifically, we show that the geometric distribution minimizes the min-entropy within the class of log-concave probability sequences with fixed variance. As an application, we obtain a discrete R\'enyi entropy power inequality in the log-concave case, which improves a result of Bobkov, Marsiglietti and Melbourne (2022).

翻译：我们使用自由程度概念的离散版本来证明对有价值的对数组合随机变量而言,自由度概念的细湿性差异性极强。更具体地说,我们显示,几何分布将有固定差异的对数组合概率序列的细湿性最小化。作为一种应用,我们在对数组合案中获得了离子R'enyi entropy权力不平等,这改善了Bobkov、Marsiglietti和墨尔本(2022年)的结果。