As Jay serveral times mentioned there is a condition that all odd derivatives are positive. This condition occured also in Szekeres paper [1] however in a slightly different context:

Definition. We call totally monotonic at if it has derivatives of any order and for every .

Then he shows that if the inverse of a function ( real analytic for , , for and , ) is totally monotonic then the regular Abel function is also totally monotonic and is uniquely determined by this property.

meets the criteria and its inverse is and is totally monotonic. Hence the regular Abel function is also totally monotonic.

In our case however the situation is a bit different. The function has no fixed point. is an Abel function for it but is not totally monotonic, but the inverse of slog is (/seems to be) totally monotonic.

I would bet there is no proof for the uniqueness claim by total monotonicity, though it sound quite plausible.

[1] G. Szekeres, Fractional iteration of exponentially growing functions, 1961.

Definition. We call totally monotonic at if it has derivatives of any order and for every .

Then he shows that if the inverse of a function ( real analytic for , , for and , ) is totally monotonic then the regular Abel function is also totally monotonic and is uniquely determined by this property.

meets the criteria and its inverse is and is totally monotonic. Hence the regular Abel function is also totally monotonic.

In our case however the situation is a bit different. The function has no fixed point. is an Abel function for it but is not totally monotonic, but the inverse of slog is (/seems to be) totally monotonic.

I would bet there is no proof for the uniqueness claim by total monotonicity, though it sound quite plausible.

[1] G. Szekeres, Fractional iteration of exponentially growing functions, 1961.