A Complex Dynamics Problem – Mathematics Stack Exchange

assume $ f (z) = frac {z} {2} + z ^ 2 $ and $ g (z) = frac {z} {2} $. I want to show that there are some $ r> $ 0 and an analytic function $ phi: mathbb {D} _r (0) to mathbb {C} $ such as $ (g circ phi) (z) = ( phi circ f) (z) $ for everyone $ z in mathbb {D} _r (0) $.

I wrote a series of power $ phi (z) = z + sum_ {n = 2} ^ infty a_nz ^ n $ and tried to solve for a recursion that the coefficients satisfy. According to the index provided, the term $ a_n $ will be expressed in terms of $ a_j, ldots, a_ {n-1} $, or $ j = lceil frac {n} {2} rceil $. Then, it should be inductively proven that for each $ n $ we have $ a_n leq16 ^ n $.

This is a special case of K├Ânig's famous linearization theorem (1884) derived from a complex dynamics.