I should add some examples of other functions that will work. Or rather, work in the same framework.
If \( h(s) \to 0 \) as \( \Re(s) \to \infty \). And \( h(s) = \mathcal{O}(|s|^{-\rho}) \) for some \( \rho > 1 \). Let's let \( h : \mathcal{H} \to \mathbb{C} \) where \( \mathcal{H} \) is some domain in \( \mathbb{C} \). Let's assume further that,
\(
\sum_{j=1}^\infty ||e^{h(s-j)}||_{\mathcal{P}} < \infty\\\
\)
For all compact subsets \( \mathcal{P}\subset\mathcal{H} \). Then the function,
\(
H(s) = \Omega_{j=1}^\infty e^{h(s-j) + z}\,\bullet z\\
\)
Satisfies,
\(
H(s+1) = e^{h(s) + H(s)}\\
\)
And,
\(
\log H(s+1) - H(s) = h(s) = \mathcal{O}(|s|^{-\rho})\\
\)
But! We can modify this idea further, by instead of taking \( h(s-j) \) and instead take \( h_j(s) \). We can only speak of these things asymptotically; with no convenient functional equation; but still valid.
So assume that \( h_j(s) : \mathcal{H} \to \mathbb{C} \). Let's assume that \( h_j(s) = \mathcal{O}(|s|^{-\rho}) \) as \( \Re(s) \to \infty \). Let's assume for all compact sets \( \mathcal{P}\subset \mathcal{H} \) that,
\(
\sum_{j=1}^\infty ||e^{h_j(s)}||_{\mathcal{P}} < \infty\\
\)
Then, the function
\(
H(s) = \Omega_{j=1}^\infty e^{h_j(s) + z}\,\bullet z\\
\)
Satisfies,
\(
\log H(s+1) - H(s) = \mathcal{O}(|s|^{-\rho})\,\,\text{as}\,\,\Re(s) \to \infty\\
\)
So again, the choice I made of,
\(
H(s) = \beta(s) = \Omega_{j=1}^\infty \frac{e^z}{e^{\displaystyle \frac{j-s}{\sqrt{1+s}}} + 1}\,\bullet z\\
\)
Is a little arbitrary; but it was chosen because the pull back will work very well.
To walk you through this. The function \( H = \beta(s) \) is defined as when \( h_j(s) = -\log(1+e^{\displaystyle \frac{j-s}{\sqrt{1+s}}}) = \mathcal{O}(e^{-|s|^{1/2}}) \) as \( \Re(s) \to \infty \). This means that,
\(
\log \beta(s+1) - \beta(s) = \mathcal{O}(e^{-|s|^{1/2}})\\
\)
Now, as I spent a lot of time proving, is that,
\(
F(s) = \log^{\circ n} \beta(s+n) \\
\)
Converges uniformly on some sector \( |\arg(s)| < \theta \) for \( \theta>0 \). And from here we want to take \( log \)'s to correct everything--make it holomorphic on \( \mathbb{C}/(-\infty,X] \) for some \( X\in\mathbb{R} \).
I'm going to summarize my proof in a couple of statements.
If \( \log(y) \) is holomorphic at \( y=1 \), and \( \log(1) = 0 \); and the branch-cut of this log is \( (-\infty,0] \); then necessarily \( \log(y) \in \mathbb{R}^+ \) for \( y\in\mathbb{R}^+ \).
Now assume that \( y(z) \) is a holomorphic function, such that \( y(1) = 1 \) and \( y(0) = 0 \). If we can define a logarithm \( \log(y) \) which is holomorphic with a branch-cut \( y \in (-\infty,0] \). Now assume further, the branch cut of \( \log(y(z)) \) can be done for \( z \in (-\infty,0] \). Then necessarily \( y(-\mathbb{R}^+) = -\mathbb{R}^+ \). Then necessarily \( y(x) \in \mathbb{R}^+ \) for \( x \in (0,1) \).
Now take our tetration function \( F \) made from \( \beta(s) \). Since \( \beta(s_0+t) \not \in \mathbb{R} \), for large \( t \) and \( s_0 \not \in \mathbb{R} \)--which I mean it isn't strictly real valued unless we're on the real line. This means, since \( F(s) = \beta(s) + \mathcal{O}(e^{-|s|^{1/2}}) \), our tetration function \( F \) is only real-valued on the real line.
This means, since \( F(s_0 + t) \not \in \mathbb{R} \) for \( \Im(s_0) > 0 \); as I mean it isn't strictly real valued over any interval, unless \( s_0 \in \mathbb{R} \). We can say that \( \log F(s_0) \neq 0 \). Because if it did, it would mean that \( F(s_0+t) \) is real-valued for \( t \) varying. BECAUSE, we can choose our branch-cuts solely dependent on the variable \( s \) in \( F(s) \). As in, if \( F(s_0) = 0 \), we can construct a holomorphic function \( F(s_0-1+s) \) which is holomorphic in \( |s| < \delta \) and \( s \not \in(-\delta,0) \).
This means we have a \( y(z) \) function which satisfies \( y(1) = 1 \) and \( y(0) = 0 \); and the branch cut of \( \log(y(z)) \) is along \( z\in(-\infty,0] \) using the principal branch of the logarithm function, with a branch cut along \( y \in (-\infty,0] \). So the function \( y(x) \in \mathbb{R}^+ \) for \( x\in(0,1) \). And therefore, \( F(s_0 + t)\in\mathbb{R}^+ \). But we know this can't happen because our tetration is only real valued on the real line.
This is essentially a mapping argument about the logarithm. Not sure how else to explain it. It's a thing, don't worry. I just may be struggling to get the point across. But the pullback isn't an issue. Again, making it work at \( \Re(s) = +\infty \) is the real problem.
If \( h(s) \to 0 \) as \( \Re(s) \to \infty \). And \( h(s) = \mathcal{O}(|s|^{-\rho}) \) for some \( \rho > 1 \). Let's let \( h : \mathcal{H} \to \mathbb{C} \) where \( \mathcal{H} \) is some domain in \( \mathbb{C} \). Let's assume further that,
\(
\sum_{j=1}^\infty ||e^{h(s-j)}||_{\mathcal{P}} < \infty\\\
\)
For all compact subsets \( \mathcal{P}\subset\mathcal{H} \). Then the function,
\(
H(s) = \Omega_{j=1}^\infty e^{h(s-j) + z}\,\bullet z\\
\)
Satisfies,
\(
H(s+1) = e^{h(s) + H(s)}\\
\)
And,
\(
\log H(s+1) - H(s) = h(s) = \mathcal{O}(|s|^{-\rho})\\
\)
But! We can modify this idea further, by instead of taking \( h(s-j) \) and instead take \( h_j(s) \). We can only speak of these things asymptotically; with no convenient functional equation; but still valid.
So assume that \( h_j(s) : \mathcal{H} \to \mathbb{C} \). Let's assume that \( h_j(s) = \mathcal{O}(|s|^{-\rho}) \) as \( \Re(s) \to \infty \). Let's assume for all compact sets \( \mathcal{P}\subset \mathcal{H} \) that,
\(
\sum_{j=1}^\infty ||e^{h_j(s)}||_{\mathcal{P}} < \infty\\
\)
Then, the function
\(
H(s) = \Omega_{j=1}^\infty e^{h_j(s) + z}\,\bullet z\\
\)
Satisfies,
\(
\log H(s+1) - H(s) = \mathcal{O}(|s|^{-\rho})\,\,\text{as}\,\,\Re(s) \to \infty\\
\)
So again, the choice I made of,
\(
H(s) = \beta(s) = \Omega_{j=1}^\infty \frac{e^z}{e^{\displaystyle \frac{j-s}{\sqrt{1+s}}} + 1}\,\bullet z\\
\)
Is a little arbitrary; but it was chosen because the pull back will work very well.
To walk you through this. The function \( H = \beta(s) \) is defined as when \( h_j(s) = -\log(1+e^{\displaystyle \frac{j-s}{\sqrt{1+s}}}) = \mathcal{O}(e^{-|s|^{1/2}}) \) as \( \Re(s) \to \infty \). This means that,
\(
\log \beta(s+1) - \beta(s) = \mathcal{O}(e^{-|s|^{1/2}})\\
\)
Now, as I spent a lot of time proving, is that,
\(
F(s) = \log^{\circ n} \beta(s+n) \\
\)
Converges uniformly on some sector \( |\arg(s)| < \theta \) for \( \theta>0 \). And from here we want to take \( log \)'s to correct everything--make it holomorphic on \( \mathbb{C}/(-\infty,X] \) for some \( X\in\mathbb{R} \).
I'm going to summarize my proof in a couple of statements.
If \( \log(y) \) is holomorphic at \( y=1 \), and \( \log(1) = 0 \); and the branch-cut of this log is \( (-\infty,0] \); then necessarily \( \log(y) \in \mathbb{R}^+ \) for \( y\in\mathbb{R}^+ \).
Now assume that \( y(z) \) is a holomorphic function, such that \( y(1) = 1 \) and \( y(0) = 0 \). If we can define a logarithm \( \log(y) \) which is holomorphic with a branch-cut \( y \in (-\infty,0] \). Now assume further, the branch cut of \( \log(y(z)) \) can be done for \( z \in (-\infty,0] \). Then necessarily \( y(-\mathbb{R}^+) = -\mathbb{R}^+ \). Then necessarily \( y(x) \in \mathbb{R}^+ \) for \( x \in (0,1) \).
Now take our tetration function \( F \) made from \( \beta(s) \). Since \( \beta(s_0+t) \not \in \mathbb{R} \), for large \( t \) and \( s_0 \not \in \mathbb{R} \)--which I mean it isn't strictly real valued unless we're on the real line. This means, since \( F(s) = \beta(s) + \mathcal{O}(e^{-|s|^{1/2}}) \), our tetration function \( F \) is only real-valued on the real line.
This means, since \( F(s_0 + t) \not \in \mathbb{R} \) for \( \Im(s_0) > 0 \); as I mean it isn't strictly real valued over any interval, unless \( s_0 \in \mathbb{R} \). We can say that \( \log F(s_0) \neq 0 \). Because if it did, it would mean that \( F(s_0+t) \) is real-valued for \( t \) varying. BECAUSE, we can choose our branch-cuts solely dependent on the variable \( s \) in \( F(s) \). As in, if \( F(s_0) = 0 \), we can construct a holomorphic function \( F(s_0-1+s) \) which is holomorphic in \( |s| < \delta \) and \( s \not \in(-\delta,0) \).
This means we have a \( y(z) \) function which satisfies \( y(1) = 1 \) and \( y(0) = 0 \); and the branch cut of \( \log(y(z)) \) is along \( z\in(-\infty,0] \) using the principal branch of the logarithm function, with a branch cut along \( y \in (-\infty,0] \). So the function \( y(x) \in \mathbb{R}^+ \) for \( x\in(0,1) \). And therefore, \( F(s_0 + t)\in\mathbb{R}^+ \). But we know this can't happen because our tetration is only real valued on the real line.
This is essentially a mapping argument about the logarithm. Not sure how else to explain it. It's a thing, don't worry. I just may be struggling to get the point across. But the pullback isn't an issue. Again, making it work at \( \Re(s) = +\infty \) is the real problem.