Toronto Math Forum
MAT3342018F => MAT334Lectures & Home Assignments => Topic started by: Jingxuan Zhang on November 21, 2018, 04:45:22 PM

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?

Definitely not.

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).

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 onetoone) itself (another very strong restriction) Fractional Linear Transforms show that the only thing you can say that $f'(0)\ne 0$ (and only because onetoone). On the other hand, if you know also (in addition to all above), that $f(0)=0$, you conclude $f'(0)=1$.