Interesting value for W, h involving phi,Omega?

[Ivars]

I was studying the graph of selfroot of Lambert function:

W(x)^(1/W(x)).

It has maximum value at x=(e*(e^e)) and is e^(1/e) ; so

W(e*(e^e)) =W(e^(e+1))= e

If we use this, we can find :

h( (e^(-(e^e)))^e))= h(1,28785E-1

ln((e^(-(e^e))))^e)= e*ln(e^(-e^e)) = -e*e^e

so :

h( (e^(-(e^e)))^e))= -W(-ln((e^(-(e^e)))^e))/ln((e^(-(e^e)))^e)=

-W(e*e^e)/-e*e^e = -e/-e*e^e = 1/e^e= e^(-e)

So:

h( (e^(-(e^e)))^e))= h(1,28785E-1= e^(-e) = 0,065988036

and second superroot of ((e^(e^e))^e) = ln(((e^(e^e))^e))/W(ln ((e^(e^e))^e)) = e*(e^e)/W(e*(e^e)) = e*(e^e)/e = e^e= 15,15426224

Ssroot( ((e^(e^e))^e)= ssroot(7,76487E+17) = e^e = 15,15426224

For these values, h(1/a) = 1/ssroot(a)

I wonder are there any similar relations for W((1/e)*(e^e)) =W(e^(e-1))= W(5.574..) = 1.3894..


Ivars