What can I say about f'(0)?

[Jingxuan Zhang]

Suppose $f:D\to\mathbb{C}$ is analytic near 0, such that $\|x\|=1\implies |f(x)|=1$. Does it follow that $f'(0)$ is purely imaginary?

[Victor Ivrii]

Definitely not.

[Jingxuan Zhang]

O.k., but is there anything I can say about $f'(0)$?

If there is really nothing to say, then please consider the following situation: $f:\mathbb{R}\to\mathbb{C}$ is continuous, and $\lim_{t\to 0} t^{-1}(f(t)+f(t)^{-1}-2)$ exists. What can say about this limit? (In particular I would love it to be 0).

[Victor Ivrii]

Basically, you cannot say anything about value of $f'(0)$. Even for analytic functions (very strong restriction), if we know that $f$ maps $\{z\colon |z|<1\}$ onto (so one-to-one) itself (another very strong restriction) Fractional Linear Transforms show that the only thing you can say that $f'(0)\ne 0$ (and only because one-to-one). On the other hand, if you know also (in addition to all above), that $f(0)=0$, you conclude $|f'(0)|=1$.