Landau inequality for several variables

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?