Fractional calculus and tetration

[fivexthethird]

My extension \( F \) is also the sole extension that is bounded by \( |F(z)| < C e^{\alpha |\Im(z)| + \rho|\Re(z)|} \) where \( \rho, \alpha, C \in \mathbb{R}^+ \) and \( \alpha < \pi/2 \).

The regular iteration for bases \( 1<b<\eta \) satisfies that, as it is periodic and bounded in the right halfplane.
What complex bases does it work for? Does it work for base eta?