I made a simple model several weeks ago which led to a conjecture, which I have been unable to prove thus far.

Problem:

Given constants and such that

,

functions and ( ) such that

and values

Prove or disprove that the following always holds:

Best attempt so far:

I've tried using other inequalities, but I always managed to find counterexamples in Mathematica.

So far, I have not found a counter examples for , so I think I'm in the right direction.

Perhaps someone knows of an application which can search for counterexamples given such conditions?

Any suggestions would be very, very, very much appreciated.