This question is from the book Complex Dynamics by Gamelin, the theorem 2.1 's proof.
Suppose $0$ is an attracting fixed point of $f$, with multiplier $\lambda$ satisfying $0<|\lambda|<1$ .
How can I get that: for $\delta$ small, there exists $C>0$, s.t.
$|f(z)-\lambda z|\leq C|z|^2, \quad |z|\leq \delta$
Absolutely, $f$ is a holomorphic function, if I use the Taylor series which means I can write $f(z)$ in this way:
$f(z)=\lambda z+ a_2z^2+\cdots + a_n z^n+\cdots$
then $|f(z)-\lambda z|=|a_2z^2+\cdots + a_n z^n+\cdots|$
due to the uncertainty of the coeffcients $\{a_n\}_{n=2}^{\infty}$
I don't know how to go on , please help me!!