Taking $\alpha$ as given, I am trying to understand/know under which restrictions on $n$, the following holds: $$ 0 \leq n+\alpha W\left(-e^{\frac{-n}{\alpha}} \right) \leq 1.$$ i.e I want to solve for $n$ the following two inequalities: a) $0 \leq n+\alpha W\left(-e^{\frac{-n}{\alpha}} \right)$, and b) $n+\alpha W\left(-e^{\frac{-n}{\alpha}} \right) \leq 1$. In addition, I know $\alpha >0$, that both $n$ and $\alpha$ are reals.
- I manage to get that a) is satisfied for $n\geq \alpha$, using:
Reduce[n+a*ProductLog[-E^(-n/a)]>=0, n, Reals]which returns(a<0&&n==a)||(a>0&&n≥a). - However I struggle to find any solution for the inequality b). I have tried all of the following:
(* for convenience I first define the function *) f[n_, a_] =n+a*ProductLog[-E^(-n/a)] Reduce[f[n, a]<=1, n, Reals] Reduce[f[n, a]<=1 && a>0, n, Reals] Solve[f[n, a]<=1, n, Reals] Solve[f[n, a]<=1 && a>0, n, Reals] Which all return:
"This system cannot be solved with the methods available to Solve/Reduce."
Even though I specified the domain as suggested here.
Is the inequality in b) really not solvable for $n$? Could Anyone think of an alternative command or commands that would do the trick?
Thank you,
EDIT:
In my context, f[n_, a_]being strictly increasing in n, and also because I know that (due to the domain of the Lambert W function) we must have n ≥ a (on the graph of f[n, a]below, we see clearly that it is not defined otherwise), even the solution for the equality to 1 would suffice me.
Manipulate[Plot[f[n, a], {n, -1, 10}], {a, 0, 5, 0.05, Appearance -> "Labeled"}] 
Solve[f[n, a] == 1, n]? $\endgroup$