Uniform Convergence of the Sequence of Compositions

• May 4th 2010, 11:21 PM
DJDorianGray
Uniform Convergence of the Sequence of Compositions
Hey everyone,
I need help with the following complex analysis problem. (It is from chapter 3 of Greene & Krantz):

Let $\displaystyle \phi : D(0,1) \mapsto D(0,1)$ be given by $\displaystyle \phi(z) = z + a_2 z^2 + a_3 z^3 + ...$ (notice in particular that $\displaystyle \phi$ is holomorphic). Let $\displaystyle \phi_1(z) = \phi(z), \phi_2(z) = \phi(\phi(z))$ and so on. Suppose that the sequence $\displaystyle \{ \phi_j \}$ converges uniformly on compact sets. Show that $\displaystyle \phi(z) = z$.

Any help is appreciated! Thank you
• May 5th 2010, 08:13 PM
Jose27
Quote:

Originally Posted by DJDorianGray
Hey everyone,
I need help with the following complex analysis problem. (It is from chapter 3 of Greene & Krantz):

Let $\displaystyle \phi : D(0,1) \mapsto D(0,1)$ be given by $\displaystyle \phi(z) = z + a_2 z^2 + a_3 z^3 + ...$ (notice in particular that $\displaystyle \phi$ is holomorphic). Let $\displaystyle \phi_1(z) = \phi(z), \phi_2(z) = \phi(\phi(z))$ and so on. Suppose that the sequence $\displaystyle \{ \phi_j \}$ converges uniformly on compact sets. Show that $\displaystyle \phi(z) = z$.

Any help is appreciated! Thank you

I haven't tried it rigourosly, but here's a naive approach that might help: If $\displaystyle \phi (z) := \sum_{j=1}^{\infty } a_jz^j$ where $\displaystyle a_1=1$, substitute $\displaystyle \varphi (z)= \lim_{n\rightarrow \infty } \phi _n(z)$ and we get $\displaystyle \phi _n(z)=\phi _{n-1}(z)+\sum_{j=2}^{\infty } a_j(\phi_{n-1}(z))^j$ and taking the limit (this is the step have to worry about) $\displaystyle \varphi (z)=\varphi (z) + \sum_{j=2}^{\infty } a_j\varphi (z)^j$ and thus $\displaystyle a_j=0$ for all $\displaystyle j\geq 2$ which gives $\displaystyle \phi (z)=z$.

PS. Great book that one, lots and lots of exercises.
• May 5th 2010, 08:33 PM
Bruno J.
What is $\displaystyle D(0,1)$?