I just realized you may not be familiar with Schwarz's lemma.
If \(f(z) : \mathbb{D} \to \mathbb{D}\) and \(f(0) =0 \); then:
\[
|f(z)| \le |f'(0)||z|
\]
In the infinite composition case; since \(f(z) : \mathbb{D} \to \mathbb{D}\); there is a unique fixed point \(z_0\) (the benefit of using a simply connected domain).
Thereby:
\[
g(z) = h(f(h^{-1}(z))\\
\]
Where \(h: \mathbb{D} \to \mathbb{D}\) biholomorphically; and sends \(z_0 \to 0\). Then:
\[
g'(0) = f'(z_0)\\
\]
The value \(h'(z_0)\) is the value of a Blashcke product (too lazy to do all the fine details); where it acts as the derivative of an automorphism. The following is an equivalent statement. If:
\[
\sum_{j=0}^\infty |f_j'(0) -1| < \infty\\
\]
Then the sum \(\sum_{j=0}^\infty |f(z) -z| < \infty\). BUT!!!! This is only true for the unit disk. If you change into a different simply connected domain; things get much more complicated; and these statements are not equivalent. You'll have to modify some steps...
This is because \(|h(f(h^{-1}(z))| \le |z| |f'(z_0)|\). Now the infinite compositions cancel out... at least to a point. The fixed points \(z_0\) can move around; but since \(f_j'(0) \to 1\), we are guaranteed \(f_j(z) \to z\), because we are guaranteed \(f\) fixes the unit disk. The unique function to satisfy \(f(z) : \mathbb{D} \to \mathbb{D}\) and \(f'(0) = 1\) is \(f(z) = z\). Therefore \(z_0 \to 0\).
And we are just checking that:
\[
0 \neq \prod_{j=1}^\infty f_j'(0) \neq \infty
\]
The quick and easy way is to just say:
\[
\sum_{j=1}^\infty |f_j'(0) -1| < \infty\\
\]
Which dates to Weierstrass...
Which is my old condition (Fereira's condition). But this is equivalent to just asking:
\[
\sum_{j=1}^\infty |f_j(z) - z| < \infty\\
\]
On the Unit Disk, Fereira's condition looks easier and nicer. But it's a very special case. And not open to generalization.
Can't stress enough that this only happens because of super nice Unit Disk behaviour, small well behaved area. This will not follow on general simply connected domains (though it'll be something similar), and will not follow on domains (Open and connected sets). But it's a great start
EDIT:
Also, since I mentioned Donald Knuth in the last post. He was the one to suggest to me, instead of writing:
\[
\Omega_{j=1}^n f_j(z) \bullet z \bullet \Omega_{j=n+1}^\infty f_j(z) \bullet z = \Omega_{j=1}^\infty f_j(z) \bullet z\\
\]
I should just write:
\[
\Omega_{j=1}^n f_j(z) \bullet \Omega_{j=n+1}^\infty f_j(z) \bullet z= \Omega_{j=1}^\infty f_j(z) \bullet z\\\\
\]
So he helped with the bullet notation !!!!!! The whole \(f \bullet g \bullet z\) was kinda his idea in some respects. I had a rough sketch; but it didn't fit right, and was clunky. He helped me stream line some shit. He really added a "functional programming" element to it, lmao.
If \(f(z) : \mathbb{D} \to \mathbb{D}\) and \(f(0) =0 \); then:
\[
|f(z)| \le |f'(0)||z|
\]
In the infinite composition case; since \(f(z) : \mathbb{D} \to \mathbb{D}\); there is a unique fixed point \(z_0\) (the benefit of using a simply connected domain).
Thereby:
\[
g(z) = h(f(h^{-1}(z))\\
\]
Where \(h: \mathbb{D} \to \mathbb{D}\) biholomorphically; and sends \(z_0 \to 0\). Then:
\[
g'(0) = f'(z_0)\\
\]
The value \(h'(z_0)\) is the value of a Blashcke product (too lazy to do all the fine details); where it acts as the derivative of an automorphism. The following is an equivalent statement. If:
\[
\sum_{j=0}^\infty |f_j'(0) -1| < \infty\\
\]
Then the sum \(\sum_{j=0}^\infty |f(z) -z| < \infty\). BUT!!!! This is only true for the unit disk. If you change into a different simply connected domain; things get much more complicated; and these statements are not equivalent. You'll have to modify some steps...
This is because \(|h(f(h^{-1}(z))| \le |z| |f'(z_0)|\). Now the infinite compositions cancel out... at least to a point. The fixed points \(z_0\) can move around; but since \(f_j'(0) \to 1\), we are guaranteed \(f_j(z) \to z\), because we are guaranteed \(f\) fixes the unit disk. The unique function to satisfy \(f(z) : \mathbb{D} \to \mathbb{D}\) and \(f'(0) = 1\) is \(f(z) = z\). Therefore \(z_0 \to 0\).
And we are just checking that:
\[
0 \neq \prod_{j=1}^\infty f_j'(0) \neq \infty
\]
The quick and easy way is to just say:
\[
\sum_{j=1}^\infty |f_j'(0) -1| < \infty\\
\]
Which dates to Weierstrass...
Which is my old condition (Fereira's condition). But this is equivalent to just asking:
\[
\sum_{j=1}^\infty |f_j(z) - z| < \infty\\
\]
On the Unit Disk, Fereira's condition looks easier and nicer. But it's a very special case. And not open to generalization.
Can't stress enough that this only happens because of super nice Unit Disk behaviour, small well behaved area. This will not follow on general simply connected domains (though it'll be something similar), and will not follow on domains (Open and connected sets). But it's a great start
EDIT:
Also, since I mentioned Donald Knuth in the last post. He was the one to suggest to me, instead of writing:
\[
\Omega_{j=1}^n f_j(z) \bullet z \bullet \Omega_{j=n+1}^\infty f_j(z) \bullet z = \Omega_{j=1}^\infty f_j(z) \bullet z\\
\]
I should just write:
\[
\Omega_{j=1}^n f_j(z) \bullet \Omega_{j=n+1}^\infty f_j(z) \bullet z= \Omega_{j=1}^\infty f_j(z) \bullet z\\\\
\]
So he helped with the bullet notation
!!!!!! The whole \(f \bullet g \bullet z\) was kinda his idea in some respects. I had a rough sketch; but it didn't fit right, and was clunky. He helped me stream line some shit. He really added a "functional programming" element to it, lmao.