Proving logarithmic inequality
I'm stuck at the following problem.
Let us have an equation that I intend to solve with some numerical method:
, where is a small number and is positive.
First off, I want to get rid of the modulus sign to avoid solving a system of equations. To do so, I need to show that
(This fact I established empirically and want to prove. The plot can be seen here: 1 is actually a limit).
The inequality is equivalent to the following one:
Denote ( ), then
And here's where I'm out of ideas. How do I analytically prove the last inequality for all ?
Any thoughts would be very appreciated!
PS I probably should have posted this in Calculus subforum, my bad.