I am looking at f(x) = (|x|+1)2. I have to find its Legendre transform, and thus I write this as
I use the fact that for x>0 and x<0 the function is strictly convex, so for these two intervals the Legendre Transform is
x>0: f*(p) = p2/4 - p
x<0: f*(p) = p2/4 + p -4
Now, for x=0 I get that f*(p) = 1 by the "usual" method.
My question is now: What is the next step? Does this mean that the function f(x) has 3 Legendre Transforms that all work for all p?
I would really appreciate any help.