After seeing the question Important formulas in combinatorics, I thought it might be of interest to have a similar list of inequalities, although not restricted to combinatorics. As with that list, there should be some rules.
- The inequality should not be too well known. This is to rule out things like Cauchy-Schwarz or the Sobolev inequalities. The inequality should be unfamiliar to a majority of mathematicians.
- The inequality should represent research level mathematics. This is taken straight from the other list, and feels like a good rule.
- The inequality should be important. Since it is easier to come up with inequalities versus exact formulas, this should be more restrictive than in the other list. The idea is to have inequalities which played an important role in the development of some field.
- An answer can be a class of inequalities. As noted in the comments, often what is important is a family of inequalities which all convey the same idea but where no single result is the fundamental example. This is perfectly acceptable, and perhaps even encouraged since any such examples will likely have lots of applications.
To give an idea of what I mean, let me give an example which I think satisfies the first three criteria; the Li-Yau estimate.
The Li-Yau inequality is the estimate $$ \Delta \ln u \geq – \frac{
n}{ 2t}.$$
Here $u: M \times \mathbb{R} \to \mathbb{R}^+ $ is a non-negative solution to the heat equation
$ \frac{\partial u}{\partial t} = \Delta u, $ $(M^n,g)$ is a compact Riemannian manifold with non-negative Ricci curvature and $\Delta$ is the Laplace-Beltrami operator.
This inequality plays a very important role in geometric analysis. It provides a differential Harnack inequality to solutions to the heat equation, which integrates out to the standard Harnack estimate.
There are many results strengthening the original inequality or adapting it to a different setting. There are also results which are not generalizations of the original inequality but which bear its influence. For instance, Hamilton proved a tensor version of the Li-Yau inequality for a manifold which has non-negative sectional curvature and evolves by Ricci flow. Furthermore, one of Perelman's important breakthroughs was to prove a version of the Hamilton-Li-Yau inequality for a solution to time-reversed heat flow when the metric evolves by Ricci flow. These results are not at all corollaries of the original Li-Yau estimate, but they are similar in spirit.
Best Answer
The class of concentration of measure inequalities is a fundamental tool in modern probability (and any field that uses probability, e.g., random matrix theory, theoretical computer science, statistics, high-dimensional geometry, combinatorics, etc.). As explained in this blog post of Scott Aaronson, these are basic ways in which one "upper bounds the probability of something bad", and often the bounds are exponential or even gaussian in nature when one is far away from the mean (or median) and there are many independent (or somewhat independent) variables involved. Examples of such inequalities include
Log-Sobolev inequalites are, strictly speaking, not concentration of measure inequalities, but are often closely related to them, thanks to techniques such as the Herbst argument.
A standard reference in the subject for these topics is
Ledoux, Michel, The concentration of measure phenomenon, Mathematical Surveys and Monographs. 89. Providence, RI: American Mathematical Society (AMS). x, 181 p. (2001). ZBL0995.60002.
I also have a blog post on this topic here.