How important are inequalities?

When reading the prefaces of many books devoted to the theory of inequalities, I found one thing repeatedly stated: Inequalities are used in all branches of mathematics. But seriously, how important are they? Having finished a standard freshman course in calculus, I have hardly ever used even the most renowned inequalities like the Cauchy-Schwarz inequality. I know that this is due to the fact that I have not yet delved into the field of more advanced mathematics, so I would like to know just how important they are. While these inequalities are usually concerned with a finite set of numbers, I guess they must be generalised to fit into subjects like analysis. Can you provide some examples to illustrate how inequalities are used in more advanced mathematics?

• Have you ever done an $\epsilon-\delta$ proof? – John Douma Apr 17 '16 at 8:16
• @JohnDouma Yes. But that only involves absolute values. By inequalities I actually mean something like the AM-GM bound. Solving basic inequalities are of course important, but what about things like Hölder's inequality, Chebyshev etc.? – Colescu Apr 17 '16 at 8:18
• All inequalities are inequal, but some are more inequal than others? – Lightness Races in Orbit Apr 17 '16 at 15:17
• All equalities are just two inequalities. – Danu Apr 17 '16 at 16:45
• @dydxx There are many good books out there. A good choice would be The Cauchy-Schwarz Master Class (which is a guide/textbook). There's a renowned book on this topic, i.e. Inequalities by Hardy and Littlewood. If you want to read books on "elementary inequalities" (without calculus or matrices etc.), you may look into books on math olympiads. – Colescu May 2 '16 at 4:16

I have a feeling that what you actually seek are examples of "famous" inequalities being put into good use and not just the notion of inequality as a general attribute.

For, if we stick to the notion of inequality in general, a prime example of why that is essential is in defining the real line.
Dedekind cuts that define the reals are partitions of the ordered field $\Bbb{Q}$. If you do not have an ordering (inequality relationships) amongst it's members, you cannot define $\Bbb{R}$ in this way.

For an example of an inequality as a "formula" , consider the $LM$-inequality, used in complex analysis, that gives an upper bound for a contour integral, thus having a variety of applications.
If f is a complex-valued, continuous function on the contour $\Gamma$, the arc length of $\Gamma$ is $l(\Gamma)=L$ and the absolute value of $f$, $|f(z)|$ is bounded by a constant $M$ $\forall$ $z$ on $\Gamma$, then it holds that $$\int_\Gamma|f(z)|dz\leq ML$$

Inequalities are extremely useful in mathematics, especially when we deal with quantities that we do not know exactly what they equate too. For example, let $p_n$ be the $n$-th prime number. We have no nice formula for $p_n$. However, we do know that $p_n \leq 2^n$. Often, one can solve a mathematical problem, by estimating an answer, rather than writing down exactly what it is. This is one way inequalities are very useful.

• For the $n$-th prime, also the asymptotics $p_n\sim n\log(n)$ is very useful, not only inequalities (in fact, $n(\ln n+\ln \ln n)>p_n>n(\ln n+\ln \ln n-1)$ for all $n\ge 6$). – Dietrich Burde Apr 17 '16 at 8:31
• @DietrichBurde Asymptotics are estimates, in a way, of a slightly different kind. I been trying to think of a good example in which you can solve an elementary question involving prime numbers by appealing to an estimate/asymptote, but otherwise would be difficult to solve. Such an example will illustrate to the OP why inequalities are so useful in this case. (By the way, I do like it how you use "log" and "ln" in the same sentence. You must be teaching calculus this semester!). – Nicolas Bourbaki Apr 17 '16 at 8:34
• Yes, I understand. Still I tend to say, that inequalities are only a small part of, say, number theory ("How important are inequalities"). – Dietrich Burde Apr 17 '16 at 8:36

There are a lot of inequalities in mathematics that are more or less important, for a list you can see here.

It is not simple to establish a rank of importance for them, but I think that the most important is the triangle inequality. In the simplest form this inequality states that, for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This captures a fundamental character of the notion of distance that agree with our intuition in the euclidean space. But it can be generalized to more abstract spaces (as the spaces of functions in functional analysis) so that, also in these spaces we can define a notion of distance.

To prove the triangle inequality in these spaces we need some other inequalities and the more relevant are the Holder and Minkowski inequalities that are used to prove when in a vector space we can define a norm, and, from this norm, a distance.

I think they're most important because of limits. I'm sure you've done limits in your calculus class. Limits are extremely important in maths - they're not just used to define derivatives and integrals.

There's a whole branch of mathematics called analysis which deals with limits. Sometime in the 18th century, mathematicians tried to understand calculus more rigorously and they came up with a formal definition of a limit.

$\lim \limits_{n \rightarrow \infty} a_n = a \iff \forall \varepsilon > 0, \exists N \in \mathbb{N} \ s.t. |a_n - a| < \varepsilon$

(In simple words, if you have a sequence of numbers say $\dfrac{1}{2}, \dfrac{3}{4}, \dfrac{7}{8}, \dfrac{15}{16}...$ which tend to a number (1 in the previous example), after a certain point ($N$) all points within a certain range ($\varepsilon$) of the limit.)

Because of this, analysis is all about inequalities - including the triangle inequality and the Cauchy-Schwartz. They're very useful.

There are other places such as in Computer Science, when you define the order of growth of an algorithm (Big-O notation), and Operations Research where you use them to put certain constraints on maximisation/minimisation problems, e.g. find the best portfolio of investments, given that you may at most invest \$1000. (The last one requires a good knowledge of theoretical probability - which is basically analysis and inequalities.)