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.