the inconsistency depending on fixpoint-selection

[Gottfried]

There is lots of other interesting stuff in Knoebel's article, but read it yourself

Yes, I've read it several times, and with the growth of my knowledge I understand increasingly more. There may still be something of whose relevance I didn't got aware yet...

For our consideration here let:
\( f_1(s)=s^{1/(s-1)} \) and \( f_2(s)=s^{s/(s-1)} \) so that \( f=f_2\circ f_1^{-1} \). So I wonder whether we can express \( f_1^{-1} \) (\( f_1 : (0,\infty)\to(1,\infty) \) and \( f_2 : (0,\infty)\to(1,\infty) \) are indeed bijective) with the Lambert W function. Any ideas?

Hmm, I'll have to study this first a bit more.