In this paper, we study derivatives of powers of Euclidean norm. We prove their Hölder continuity and establish explicit expressions for the corresponding constants. We show that these constants are optimal for odd derivatives and at most two times suboptimal for the even ones. In the particular case of integer powers, when the Hölder continuity transforms into the Lipschitz continuity, we improve this result and obtain the optimal constants.

中文翻译：

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欧几里得范数幂的平滑度参数

在本文中，我们研究了欧几里得范数的幂的导数。我们证明了它们的 Hölder 连续性，并为相应的常数建立了明确的表达式。我们表明，这些常数对于奇数导数是最优的，而对于偶数导数最多是次优的两倍。在整数幂的特殊情况下，当 Hölder 连续性转换为 Lipschitz 连续性时，我们改进了这个结果并获得了最优常数。