tetration bending uniqueness ?

[bo198214]

Hmm. This makes me wonder about the following conjecture:

The "principal" analytic fractional iterates \( \exp^t(x) \), \( t \ge 0 \) of the natural exponential (and perhaps any with \( b > \eta \)) are uniquely characterized by

\( \frac{d^n}{dx^n} \exp^t(x) > 0 \) for all \( x \), all \( t \ge 0 \) and all \( n > 0 \).

There is also an associated line of thought of Szekeres, but not with the fractional iterates but with the Abel function. He wonders about the alternating signs in the logarithm and shows that the principal/regular Abel function of \( e^x-1 \) is uniquely determined by the "totally monotonic"-criterion.