I have recently seen questions involving bizarre inequalities, usually consisting of cycling over variables; here's one example (see also related links): $$\sum\limits_{cyc}\frac{1}{\sqrt{2a^2+5ab+2b^2}} \geq\sqrt{\frac{3}{ab+ac+bc}}$$ These questions are often highly upvoted. I struggle to see the interest in this.
Q: What is the interest/application of this area of mathematics?
In attempting to investigate this myself, I found this three-line description on Wikipedia discussing the "AM-GM inequality", which many of the questions use:
An important practical application in financial mathematics is to computing the rate of return: the annualized return, computed via the geometric mean, is less than the average annual return, computed by the arithmetic mean (or equal if all returns are equal). This is important in analyzing investments, as the average return overstates the cumulative effect.
Are there any other applications?
Edit: I should have included what @leslietownes had to say in chat:
for whatever reason, such problems seem to be very popular in contests. outside of 'contest math' i do not know of any reason to care. one desirable quality of a contest problem is that it be doable without advanced knowledge (particular, it ought to have at least one way of solving it that is both short and 'low tech'). another desirable quality is that a problem not be "too easy." so, random inequalities where you need to deploy a number of non-obvious but low tech 'tricks' fit the bill.
