Yes, I see - the area is either infinite or zero. One may think of taking the area in some finite boundaries - but then we may be able to construct a function, which produces a lower area in that region (compensated by a relatively bigger area outside that region) So I see, this is no conclusive solution, yet.
The only idea that remains is, to consider the length of the guessed curve of f°0.5(x) in an interval of f°0.5(x) .. f°2.5(x). (but also... I've never dealt with line-integrals). This length should be minimal if the amplitude of the overlaid mod(1)-function is minimal. But this is just crude guess - should be checked on paper first...
Hmmm.
[update]
Just added two more pictures.
The first gives mirrored graphs, which actually means the functional inverses f°-1(x) are also shown.
Then we can compute distances for each function and its inverse (the distances are measured along the antidiagonal direction). The "smoothest" version of f°0.5(x) has a "smooth" graph for the distances; if all derivatives are monotonuous only of that selected curve then... we have a special case...
The red line shows the distances of the "smoothest", the orange line some other, not optimal, estimate for f°0.5(x)
[/update]
Gottfried
[update2]
Here is the "distances"-view where the "mirror"-graphs is just rotated by 45 deg. Also a third interpolation was included. Perhaps the optimal view of the matter
[/update2]
The only idea that remains is, to consider the length of the guessed curve of f°0.5(x) in an interval of f°0.5(x) .. f°2.5(x). (but also... I've never dealt with line-integrals). This length should be minimal if the amplitude of the overlaid mod(1)-function is minimal. But this is just crude guess - should be checked on paper first...
Hmmm.
[update]
Just added two more pictures.
The first gives mirrored graphs, which actually means the functional inverses f°-1(x) are also shown.
Then we can compute distances for each function and its inverse (the distances are measured along the antidiagonal direction). The "smoothest" version of f°0.5(x) has a "smooth" graph for the distances; if all derivatives are monotonuous only of that selected curve then... we have a special case...
The red line shows the distances of the "smoothest", the orange line some other, not optimal, estimate for f°0.5(x)
[/update]
Gottfried
[update2]
Here is the "distances"-view where the "mirror"-graphs is just rotated by 45 deg. Also a third interpolation was included. Perhaps the optimal view of the matter
[/update2]
Gottfried Helms, Kassel