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There exists a dimensional constant $C_n$ such that, for each holomorphic function $f:\overline{B(1)}\to \mathbb{C}$ on the closed unit ball centered at the origin of $\mathbb{C}^n$ and each multi-index $\alpha=(\alpha_1,\ldots, \alpha_n)$, $$\frac{1}{\alpha!}\left|\frac{\partial^{|\alpha|f}}{\partial z^\alpha}(0)\right|\le C_n^{|\alpha|} \sup_{\overline{B(1)} } |f|.$$ Here $|\alpha| := \alpha_1+ \cdots + \alpha_n$, $z^\alpha = z_1^{\alpha_1} \cdots z_n^{\alpha_n}$ and $\alpha! = \alpha_1! \cdots \alpha_n!$.

Indeed $C_n$ can be taken to be $\sqrt{n}$. This is because the closed polydisc $\overline{\Delta(1/\sqrt{n})} := \{z\in \mathbb{C}^n: \, |z_1|,\ldots, |z_n| < 1/\sqrt{n}\}$ is contained in the closed unit ball $\overline{B(1)}$, so that, by the multivariate Cauchy integral formula for derivatives, $$\frac{1}{\alpha!}\left|\frac{\partial^{|\alpha|f}}{\partial z^\alpha}(0)\right|\le \frac{\sup_{\overline{\Delta(1/\sqrt{n})} } |f|}{(1/\sqrt{n})^{|\alpha|}} = \sqrt{n}^{|\alpha|} \sup_{\overline{\Delta(1/\sqrt{n})} } |f| \le \sqrt{n}^{|\alpha|} \sup_{\overline{B(1)} } |f|.$$

Can the dimensional constant $C_n$ be improved?

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    $\begingroup$ Can you provide a reference for the inequality in the first display? Guillaume $\endgroup$
    – Guillaume
    Jul 20, 2015 at 15:29

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