Let $ f : U \to \Bbb C$ be a holomorphic function in a neighborhood of 0 (or a polynomial). Is it true that for any integer $ k \geq 1$ , $ $ |f^{(k)}(0)| \leq \|D^k_0 |f|\| \left( = \max_{|u_1|=\dotsb = |u_k|=1} \| D^k_0 |f| (u_1,\dotsc,u_k)\| \right)? $ $ where $ D^k_0 |f|$ is the $ k$ th derivative at zero of the map $ |f| : \Bbb C \simeq \Bbb R^2 \to \Bbb R$ .

For $ k=1$ there is an equality. For $ k=2$ , the inequality holds true (I computed it, but I have no easy way to prove it).

It holds that $ \| D^k_0 \log |f|\| = \left|\frac{\partial^k}{\partial z^k} \log f(z)\right|$ (because $ \log |f| = \Re(\log f)$ ) but I can’t obtain anything out of that.

I tried also to exploit the formula $ f^{(k)}(0) = \frac{1}{2\pi i}\oint z^{-k-1} f(z) dz$ but without success.

Any idea?