Universal uniqueness criterion?

[bo198214]

Consider following Assumption:
There exist entire 1-periodic function \( h \) such that
\( G(z)=F(z+h(z)) \) is also analytic tetration on base \( b \).

Then, function \( I(z)=z+h(z) \) is not allowed to take values -2, -3, ..
being evaluated at elements of \( \mathbb{C}^{\prime} \).

But \( I(z) \) does not need to be entire, it can also omit the values -2, -3 , ... as arguments.
For an entire function we would know that it can omit at most 1 value (Little Picard).

Edit: Oh I see now \( I(z) \) needs not to be entire, but can be continued being entire by \( I(z\pm n)=I(z)\pm n \) if \( I=f^{-1}\circ g \) can be/is defined on the strip \( S \).