I have the following function

where and are constants and where is a monotonically

decreasing function of ( stands for absolute value of derivative). According to my model, all

signs indicate that is a monotonically increasing function of

. Numerical experiments and logical arguments confirm this but

I need a rigorous proof of this. If is not unconditionally monotonically

increasing function of I wish to know under what conditions it

will be monotonically increasing function of .