I would like to find the range of the function below, but only for values of $a$ strictly greater than zero.
v(a_)=a*Log(1(a  a E^(1  1/a + ProductLog(E^(1 + 1/a))))/a)
In other words: what is the range of $v(a)$, for/conditional on $a>0$ (reals).

I have tried this: (ie without restricting)
FunctionRange(v(a), a, y)
, but the computation time is too long, I can’t manage to get an output. 
I also have tried to define the $v(a)$ function over the domain that interests me using this:
v(a_ /; 0 < a) := a*Log(1(a  a E^(1  1/a + ProductLog(E^(1 + 1/a))))/a)
(solution found here), and thenFunctionRange(v(a), a, y)
, but the computation time seems to bee stil too long. Probably due to the fact that my restricted definition doesn’t seem to have worked: even in that case,FunctionDomain(v(a),a)
returns $a≠0$ (instead of $a>0$ as I would have expected since it is what I defined in the first place).
EDIT
 This doesn’t seem to work either:
Assuming(a>0,FunctionRange(v(a), a, y))
.