If I have two analytic complex functions $f_1$ and $f_2$ which each conformally map the unit disc to regions $\Omega_1$ and $\Omega_2$ respectively, where $\Omega_1\subset\Omega_2$, such that $f_1(0)=f_2(0)$, what can I say about the relationship between the moduli of the derivatives at $0$? It seems like it'd be necessary for $|f_2'(0)|$ to be larger to compensate for the difference in range size. Is it possible for that relationship to switch somewhere else in the disc (i.e. $|f_2'(z_0)|\le|f_1'(z_0)|$ for some $z_0\ne 0$ in the unit disc) while still preserving the containment configuration of the ranges? I suspect that conformality prevents that from being possible, and that it may be the mechanism which allows the initial growth rate of the functions to dictate their future growth.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
Here's how it works:
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top