For $f \in C^n(\mathbb{R})$ and $0 < \alpha < n$, Landau-Kolmogorov inequlity is geven by $$ \|f^{(\alpha)}\| \leq K(n,\alpha)\| f\|^{1-\alpha/n}\|f^{(n)}\|^{\alpha/n}, 0 < \alpha < n,$$ where $\| \cdot \|$ is the sup norm and $K(n,\alpha)$ is a constant depend on $n $ and $ \alpha$. Is there similar inequlity for function $f \in C^n(\mathbb{R}^m)$ ?

In this paper, https://core.ac.uk/download/pdf/82427932.pdf Page 323 mentioned such an inequality with a specific constant $\pi/2$. Precisely it says if $g\in C^k(\mathbb{R}^n), |D^{k}|\leq K, $ and $ |g|\leq M, $ then $$ |D^{\alpha}g| \leq \frac{\pi}{2}M\left(\frac{K}{M}\right)^{\alpha/k}, \alpha = 0,1, \dots , k. $$ There is no justification for this inequlity. but i think this is a generalaztion of Landau inequlity.

Can some one help me with the justification or with a reference?


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