Lets summarize what we have so far:

Proposition. Let be a vertical strip somewhat wider than , i.e. for some and .

Let for some and let be two domains (open and connected) for values, and let be holomorphic on . Then there is at most one function that satisifies

(1) is holomorphic on and

(2) is real and strictly increasing on

(3) for all and

(4) There exists an inverse holomorphic function on , i.e. a holomorphic function such that for all .

Proof. Let be two function that satisfy the above conditions. Then the function is holomorphic on (because and (4)) and satisfies . By (3) and (4)

and .

So can be continued from to an entire function and is real and strictly increasing on the real axis.

by our previous considerations.

By Big Picard every real value of is taken on infinitely often if is not a polynomial, but every real value is only taken on once on the real axis, thatswhy still . But this is in contradiction to . So must be a polynomial that takes on every real value at most once. This is only possible for with because ..

In the case of tetration one surely would chose and or . However I am not sure about the domain which must contain and hence give some bijection , with some .

Of course in the simplest case one just chooses if one has some function in mind already. However then we can have a different function with but our intention was to have a criterion that singles out other solutions.

So we need an area on which every slog should be defined at least and satisfy as well as .

Proposition. Let be a vertical strip somewhat wider than , i.e. for some and .

Let for some and let be two domains (open and connected) for values, and let be holomorphic on . Then there is at most one function that satisifies

(1) is holomorphic on and

(2) is real and strictly increasing on

(3) for all and

(4) There exists an inverse holomorphic function on , i.e. a holomorphic function such that for all .

Proof. Let be two function that satisfy the above conditions. Then the function is holomorphic on (because and (4)) and satisfies . By (3) and (4)

and .

So can be continued from to an entire function and is real and strictly increasing on the real axis.

by our previous considerations.

By Big Picard every real value of is taken on infinitely often if is not a polynomial, but every real value is only taken on once on the real axis, thatswhy still . But this is in contradiction to . So must be a polynomial that takes on every real value at most once. This is only possible for with because ..

In the case of tetration one surely would chose and or . However I am not sure about the domain which must contain and hence give some bijection , with some .

Of course in the simplest case one just chooses if one has some function in mind already. However then we can have a different function with but our intention was to have a criterion that singles out other solutions.

So we need an area on which every slog should be defined at least and satisfy as well as .