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 \( f(x) \) totally monotonic at \( x_0 \) if it has derivatives of any order and \( (-1)^{k+1}f^{(k)}(x_0)>0 \) for every \( k>0 \).
Then he shows that if the inverse of a function \( f \) (\( f \) real analytic for \( x\ge 0 \), \( f(x)>x \), \( f'(x)>0 \) for \( x>0 \) and \( f(x)=x+ax^2+\dots \), \( a>0 \)) is totally monotonic then the regular Abel function is also totally monotonic and is uniquely determined by this property.
\( x\mapsto e^x-1 \) meets the criteria and its inverse is \( x\mapsto \log(x+1) \) 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 \( e^x \) has no fixed point. \( \text{slog}_e \) 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 \( f(x) \) totally monotonic at \( x_0 \) if it has derivatives of any order and \( (-1)^{k+1}f^{(k)}(x_0)>0 \) for every \( k>0 \).
Then he shows that if the inverse of a function \( f \) (\( f \) real analytic for \( x\ge 0 \), \( f(x)>x \), \( f'(x)>0 \) for \( x>0 \) and \( f(x)=x+ax^2+\dots \), \( a>0 \)) is totally monotonic then the regular Abel function is also totally monotonic and is uniquely determined by this property.
\( x\mapsto e^x-1 \) meets the criteria and its inverse is \( x\mapsto \log(x+1) \) 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 \( e^x \) has no fixed point. \( \text{slog}_e \) 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.