# List of inequalities including proofs

Many questions I have seen involve inequalities, and searching on the net, I couldn't find lists of common inequalities that included proofs (or, even better, links to proofs).

Why isn't there such a list here? (Or is it, and I only missed it?)

I guess it would be quite easy and terribly useful to have such a thing, just like a specialized FAQ. Content could be provided from so many Q&A with only a little rewriting.

Edit: what I have in mind is a collection of inequalities, as formulas, and, if possible, with a name (like "John Smith's inequality). All on one page and when I click on any of those, I'd like to get to a page with a proof. Nothing I have seen on the net comes even close to that.

Oh, well, never mind, I repent and delete.

• Somewhat related Wikipedia pages: List of inequalities and Category:Inequalities. – Martin Sleziak Jan 22 '16 at 18:41
• @MartinSleziak those are lacking proofs and cannot be easily checked (formulas not on one page). What's more, I don't trust wikipedia to be the same tomorrow. Wikipedia is goog enough if I want to look up "John Smith's inequality", but fails when I'm looking for something involving $\ln(1+x^2)$. – Gyro Gearloose Jan 22 '16 at 18:50
• To which extent this could be useful is disputable, but I pretty much doubt that creating good list of frequent/useful inequalities would be easy. However, creating lists based on Math.SE questions is not without precedent here. See Catalog of standard exercises and Would “organizer posts” be useful/welcome here?. – Martin Sleziak Jan 22 '16 at 18:51
• To respond to your inquiry about searching for ineqs with $\ln(1+x^2)$, both using Google, maybe with restricting it to a specific site, and using tags do not seem like terrible options. For more tips on searching see here and in other posts linked there. – Martin Sleziak Jan 22 '16 at 18:54
• @MartinSleziak I never trusted Google to be the same tomorrow. And what I found using google either had no proofs or were articles with proof but only for a specific inequality. – Gyro Gearloose Jan 22 '16 at 19:01

This is a site for questions and answers. A list of inequalities is neither a question nor an answer. This is why it's not here, and should not be.

• Looking for an inequality is kind of a question. – Gyro Gearloose Jan 22 '16 at 19:06

Let me try to state more clearly some things which I already have mentioned in comments.

There are some lists of results which have links to posts on math.SE. So these lists could be somewhat similar to what you have in mind. But notice that they were made mainly to avoid duplication of questions which are asked on this site very often, so the purpose is somewhat different than what you are asking about. Another thing worth mentioning is that rather comprehensive list of limits (see the first link below) is not hosted here on math.SE, but on a separate site. Links to some relevant older meta discussions:

The reason why I think that creating something similar for inequalities would not be feasible is that it is very broad topic. If the intention is to have some list which is short enough to be posted as an answer on this site, it should probably be more focused. (Of course, I could be wrong.)

The question asking where such list can be found could have better chance to be on topic here. For example, this question was quite well received: A comprehensive list of binomial identities? In fact, some questions asking about books concerning inequalities have been posted on this site before, see Books for inequality proofs and Is there any book about inequality? (This is not the same thing as what you want, but it is probably still worth mentioning.)

Sounds good, but please organize it as one answer per type of inequality instead of a million answers each with one named inequality.

• Depending on what concept you have in mind about a type, I'm quite with you. $\ln(1+x)\leq0$ and $\ln(1+y)\leq0$ should clearly be handled as one case. I'm advocating, apart from this, a standardized formalism, (not enforced, but with some preference), for example to use $\ln$ instead of $\log$ in case the natural logarithm is used. And to use $x$ for the first real variable, and $z$ for the first complex variable. These examples should be quite acceptable by now, and a normalized, but still open language would be a great benefit. – Gyro Gearloose Jan 28 '16 at 20:38
• Categories such as 3-variable polynomial inequalities, norms on vector spaces, distance functions, symmetric function inequalities, ...., each of which contain many inequalities. – zyx Jan 28 '16 at 23:59