Creating this inequality I was thinking to Claude Leibovici who loves Lambert's function .So we have :
Let $e^{-1}\leq c\leq b\leq a\leq 1 $ such that $|\ln(abc)|=a+b+c$ then we have : $$3\Omega^2\leq ab+bc+ca<1$$ Where $\Omega$ denotes the Omega constant .
First I have tried to use the Am-Gm inequality unsuccessfully.We can delete a variable using Lambert's function(playing with the condition).And after I have tried to use derivatives to conclude but there is some steps that I cannot follow.
Edit :
I link the page on Lambert's function https://en.wikipedia.org/wiki/Lambert_W_function .Maybe someone can find an idea to prove it.
If you have some ideas...
...Thanks a lot .
Ps:Can someone explain the down-vote ?