Standard techniques for proving inequalities Recently, I came across the following problem:
Let $S = \sum_{i=1}^{n} a_i$ where $a_i's \geq 0$. Then prove the following inequality: $$\sum_{i=1}^{n} \frac{S}{S-a_i} \geq \frac{n^2}{n-1}$$
and after trying various other methods(which failed) I came across the AM - HM inequality which made solving the problem very easy. Later I have seen AM - GM inequality and tweaking values I found these two interesting bounds on the area of a triangle.
(i) Area $\leq \frac{S^2}{4}$
(ii) Area $\leq \frac{S^2}{3 \sqrt3}$ , where $S$ is the semi perimeter of the triangle.
Can you give an example of your favorite inequality using which simplified the entire problem ?
 A: Yes and no, there are so many results about inequalities that it is imposible to answer your question. Depending on what are you studying, the technics will be different.
For example in real analyis you might find 'handy' the AM-GM inequality or Cauchy-Schwarz inequality, or maybe formulas like $ \mid \sin(x) \mid\leq1$. In fourier analysis you will find useful formulas like Bessel's inequality or Passerval's inequality. In functional analysis it is frecuent to use Hölder and Minkowski inequalities,...
That is why I think this question has no answer, if you are working in a small field of maths you will have a bunch of 'handy' inequalities that will appear quite often, but that will be of no help in other fields. And I have talked about anaysis, but inequalities apears in almost every field of maths like probability, abstract algebra, numerical analysis, differential equations,...
So yes, there are a lot of useful inequalities, but depends on what are you going to study and at which level of depth. For example, I am doing a PhD in group theory, so I dont think I'll ever use a Sovolev inequality in my reserach, but a PhD in differential equations must know these, since it is of vital importance in its field.
