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 \subseteq G\subseteq f(D)=G')
(2)
is real and strictly increasing on 
(3)
for all
and =y_1)
(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
Let
(1)
(2)
(3)
(4) There exists an inverse holomorphic function
Proof. Let
So
By Big Picard every real value of
In the case of tetration one surely would chose
Of course in the simplest case one just chooses
So we need an area