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
\( \alpha \uparrow^2 z = L - \sum_{n=1}^\infty a_n \lambda^{nz} \)
where
\( \alpha^L = L \) and \( \lambda = \log(L) \)
If \( h(\lambda z) = \alpha^{h(z)} \) is the inverse schroder function, normalized so that \( h'(0) = 1 \), then
\( a_n = -h^{(n)}(0) h^{-1}(1)^n \)
where
\( h^{-1}(1)^n(-1)^n > 0 \)
and, as Sheldon mentions, but I haven't seen a fully rigorous proof,
\( h^{(n)}(0)(-1)^{n+1} > 0 \) for \( n \ge 1 \)
therefore \( a_n>0 \). Now it follows that
\( |\alpha \uparrow^2 z - L| \le \sum_{n=1}^\infty a_n \lambda^{nx} \le \sum_{n=1}^\infty a_n = L -1 \)
and tetration is contained in a disk about \( L \) that lies in the right half plane.
NOW, all I need is \( h^{(n)}(0)(-1)^{n+1} > 0 \) for \( n \ge 1 \). 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
\( \alpha \uparrow^2 z = L - \sum_{n=1}^\infty a_n \lambda^{nz} \)
where
\( \alpha^L = L \) and \( \lambda = \log(L) \)
If \( h(\lambda z) = \alpha^{h(z)} \) is the inverse schroder function, normalized so that \( h'(0) = 1 \), then
\( a_n = -h^{(n)}(0) h^{-1}(1)^n \)
where
\( h^{-1}(1)^n(-1)^n > 0 \)
and, as Sheldon mentions, but I haven't seen a fully rigorous proof,
\( h^{(n)}(0)(-1)^{n+1} > 0 \) for \( n \ge 1 \)
therefore \( a_n>0 \). Now it follows that
\( |\alpha \uparrow^2 z - L| \le \sum_{n=1}^\infty a_n \lambda^{nx} \le \sum_{n=1}^\infty a_n = L -1 \)
and tetration is contained in a disk about \( L \) that lies in the right half plane.
NOW, all I need is \( h^{(n)}(0)(-1)^{n+1} > 0 \) for \( n \ge 1 \). 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).