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 \( n > 0 \).

You should put that in the open problems section. Really interesting question. And you can even put \( n\ge 0 \).