Also if you apply ln(P) as utility function then the partial derivate for W yields the same result to U(P) = U(AW) as to U(P) = U(W), namely 1/W. So in that case your inequality does not hold.
[U(G(W))]' = U'(G(W))G'(W)
[ln(AW)] ' = ln'(AW) [AW]' = (1/(AW)) A = 1/W
So in that case [U(P)=ln(P)] your inequality does not hold.
The inequality I gave is a criteria, not a theorem. I didn't claim it was always satisfied, I just claimed it was possible.
If it is satisfied (e.g., U(P)=sqrt(P)), a NIT creates an incentive for work. If equality holds (U(P)=ln(P)), a NIT has no marginal effect on work. If the reverse inequality holds (U(P)=1-1/P), a NIT creates a disincentive for work.