Of course, as soon as I post this I've essentially proved it using the Fourier expansion. Now I'm just hoping Sheldon comes on and can enlighten me a bit more as to why this works.
First and foremost
where
and )
If = \alpha^{h(z)})
is the inverse schroder function, normalized so that  = 1)
, then
}(0) h^{-1}(1)^n)
where
^n(-1)^n > 0)
and, as Sheldon mentions, but I haven't seen a fully rigorous proof,
}(0)(-1)^{n+1} > 0)
for 
therefore
. Now it follows that
and tetration is contained in a disk about
that lies in the right half plane.
NOW, all I need is for . Sheldon has claimed this, but in order to put it in the paper, I'll probably either need to prove it myself, or have a good reference. In a short enough, well thought out manner, that is rigorous enough to put in the finalized version of my paper (credit will be given, of course).
First and foremost
where
If
where
and, as Sheldon mentions, but I haven't seen a fully rigorous proof,
therefore
and tetration is contained in a disk about
NOW, all I need is