Well, I'm learning olympiad inequalities, and most of the books go like this:
Memorize holder-cauchy-jensen-muirhead, then just dumbass (i.e bash and homogenize) and use the above theorems.
Needless to say, I feel extremely disintersted in inequalities for this approach.
What are some good books which makes inequality interesting ? (Not involving extremely elementary things like AM-GM, Cauchy, but more advanced like Jensen-Muirhead-Karamata-Schur also) ?
To give clarification what I consider as intersting as of now, the only inequality problem that I found interseting is that the following statement for positive reals
$\displaystyle \sum_{i=0}^{n}\frac{a_i}{a_{i-1}+a_{[i+1 \mod n}]} \leq \frac{1}{2}$
only holds for finitely many $n$. Also one thing that I find interesting is that the bounds which can't be analytically found, but exists.