I am doing an exercice to prove the convergence of the Newton's Method.
Let f b an holomorphic function on a simply connected space.
$\ x^* \epsilon \space \Omega, \space f(x^*) = 0 \space $and $\ \space f'(x^*)\neq 0 $.
$\ g = id - \frac{f}{f'} $ with id the identity function.
I proved that $\ g'(x^*) = 0 $
Now I have ton conclude that there exist a circle centered in $\ x^* $ with a radius $\ \alpha \neq 0$ such as $\ |g'(z)| < \epsilon $. (with $\ \epsilon < 1 $)
I don't know where to begin.