The semi-group iso problem and bounded derivatives

[tommy1729]

A nice exercise is perhaps this

Let f(f(x)) = exp(x)

for all x :

f ' (x) > 0

f ' ' (x) > 0

what is the lowest possible value of A ;

A = inf f(x)

or equivalent 

A = f( - oo )

Does A have a closed form ?

That is a nice idea.

***

Following the above

consider the family of functions f_r(x) such that :

f_r^[r](x) = exp(x) 

with r between 0 and 1.

And consider 

for all x :

f_r ' (x) > 0

f_r ' ' (x) > 0

what are the lowest possible values of A( r ) ;

A( r ) = inf f_r(x)

or equivalent 

A( r ) = f_r( - oo )

It is assumed that the lowest possible value A®  for a given r is independant of the lowest possible values for other values of a different r.


This implies that there is a unique lowest function A of the variable r without contradictions : A( r ).

A( r ) = f_r(- oo) = sexp( slog(-oo) + r)

f_r ' (x) = d/dx sexp( slog(x) + r)

f_r ' ' (x) = d^2/d^2x sexp( slog(x) + r)

 
We then have a uniqueness condition ; A® uniquely defines sexp resp. slog by the above ( and the trivial sexp(x+1) = exp(sexp(x)) )

Food for thought.



regards

tommy1729