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Is it known?

If $f:D→\mathbb{C}$ is holomorphic function in $D$ , where $D$ is compact, connected (and convex, if necessary) set, and \begin{align} |f^{(k)}| \ge 1 \ \ \text{ on } \ D \end{align} holds then \begin{align} |\{z \in D : |f|\le \alpha\}| ≤ C\alpha^{2/k} \end{align} (Of course, Real version is well-known.)

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