I have the following question that I am currently unable to satisfactorily answer myself.
My question is:
Does the inequality
$$\frac{a}{b} + \frac{b}{a} < \frac{f(a)}{f(b)} + \frac{f(b)}{f(a)}\tag{*}$$
imply that $a < b$, in general (i.e., for ALL functions $f$)?
If the answer to my question is NO, under what conditions on the function $f$ is it true that inequality (*) implies $b < a$?
The given inequality is symmetric in $a$ and $b$. So if it holds for some pair $(a,b)$, it also holds for the pair $(b,a)$. Thus it cannot force $a\lt b$.