Let $x > -2$ and $f(0) = 1,f(x+1) = \exp(f(x))$
And $f$ is a taylor series :
$$f(x) = \sum_n a_n x^n$$
where the $a_n$ are all real and all nonzero and $f$ has radius $2$.
(Notice $f(-1) = 0, f(-2) = \ln(0)$. The functional equation holds for all $x>-2$. So the radius is at most $2$.)
(There are infinitely many solutions now but we talk about adding a condition below..)
Let $A$ be a real number ; the infimum of the sum of $a_n^2$. It is thus a condition on the reals $a_n$.
see : https://en.wikipedia.org/wiki/Infimum_and_supremum
$$ A = \inf \sum_n a_n^2 $$
Do we have uniqueness of $f$ here ? In other words is there a unique $f$ that minimizes $\sum_n a_n^2 $ ?
Bonus question : If we add the condition $f ‘(x) > 0 $ for all $ x >0$ to the above , do we get uniqueness for $f$ then ?
I thought about calculus of variations, but my understanding is limited.