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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 $b < a$?

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@André Nicolas, please see my latest comment below your answer, which I have already accepted - thank you! :) – user11235813 Feb 16 '13 at 21:49
@André Nicolas, would you be kind enough as to shoot me an e-mail, as I could not determine your e-mail address from a quick web search? My e-mail address is contained in my paper in JIS. Thanks! (I just want to show you what this question is for, and what I have done so far...) – user11235813 Feb 16 '13 at 22:01
up vote 5 down vote accepted

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 $a\lt b$.

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And to answer the question, the given inequality will never imply $a<b$, unless the inequality is absurd (i.e., if holds for no pair $(a,b)$ at all; ex falso sequitur quodlibet). – Marc van Leeuwen Feb 14 '13 at 5:27
Thanks Andrei, thanks Marc! – user11235813 Feb 14 '13 at 5:31
@André Nicolas, just a quick follow-up to your answer: How about if we have the added condition that $\frac{f(a)}{a} < \frac{f(b)}{b}$? Note that this last inequality is no longer symmetric. – user11235813 Feb 16 '13 at 21:48

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