Which (if any) inequalities with real numbers should have separate tags?

Question

There are certainly many inequalities which are rather important and useful and which appear frequently in various areas in mathematics (AM-GM, Jensen, Cauchy-Schwarz, etc.) The question I want to ask is whether some of them would be also useful as tags on this site. And if yes, for which of them tags should be created.

To avoid making this question too general, let's deal just with the inequalities concerning real numbers. So for the purpose of this question let us leave inequalities from probability (such as Chebyshev's inequality), analysis (Harnack's inequality, Grönwall's inequality, etc.) and various other areas aside.

If we decide that some the inequalities should have their own tags, we should also discuss when these tags are supposed to be used. Are they only for the questions about these inequalities? Should they be added to the questions where the OP explicitly says that they want to see solution using this particular inequality? Or if some an answer is given which uses some inequality, should the tag be added based on the answer?

One more thing to keep in mind is that question can have at most five tags. So if we create too many very specific tags, it might happen that on some question we will not have enough space to add all tags which might be suitable there.

On the other hand, I can see that inequalities tag might start to resemble a big monolith of questions which are not divided into subcategories, which make questions about inequalities rather difficult to search. (Although I am not sure to which extent they can be reasonably categorized. And whether tags could actually improve searching among these question, or whether it is simpler to search for the exact formula.) It would be great if some of the users who are active in the inequality tag could comment on whether they think that inequalities tag needs to become more organized.

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The reason why I have decided to post the question now was that not so long ago two tags of this nature have been created. Namely a.m.-g.m.-inequality was created in January and young-inequality created very recently. (Another similar tag which existed, although only for a very brief period, was cauchy-schwarz.)

In both cases the tags start growing relatively fast. Therefore I considered asking about community opinion as a reasonable thing to do. If the community consensus is that those tags should be removed, it is better to find out before the tags contain too many questions and removing the tags is a lot of work. (Although moderators can remove tags without bumping - see here and here - but this only works if there is another tag where all currently tagged question would fit. Possibly in the case of these tags, inequalities might be a good fit for most questions, so it is possible that in this case we will not have to do manual retagging of all questions if the tags are removed.)

I have previously asked about a.m.-g.m.-inequality in the tag management thread. But since some other similar tags started appearing, it is probably better to discuss them in general.

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EDIT: Recently (end of July 2017) two new inequality-related tags were created, namely cauchy-schwarz-inequality and holder-inequality. The list of questions currently having these tags can be found here.

EDIT2: And the tags rearrangement-inequality, jensen-inequality, convexity-inequality, muirhead-inequality have been created a few days later.

EDIT3: The tags karamata-inequality and tangent-line-method were added today. The latter is a method for proving inequalities rather than a specific inequality - the same is true about the tags mentioned in Jyrki's answer. This kind of moved the question a bit into contest-math.

I guess the above edits show that number of inequality-related tags is still growing - so I will not post further updates when new such tags appear. (I guess the above are sufficient as examples.)

Answer 1

I think all of those that you mentioned (AM-GM, Jensen, Cauchy-Schwarz) are just fine for tags.
I would even say Chebyshev's and Markov's inequalities are fine for tagging, as well as even the union bound, though not the other two you mentioned.

My rule of thumb would be that an inequality deserves a tag if:

• Mathematically-inclined students would learn it before college (e.g. AM-GM, triangle), or

• It is very likely to be taught in advanced undergraduate or introductory graduate courses outside of a math/physic/statistics department (e.g. Jensen's, Cauchy-Schwarz, Chebyshev)

The rationale would be that the audience would likely be large enough in these scenarios that the tags are very likely to flourish and help the question get better views.

Answer 2

Drawing attention to a couple more recently created related tags. Namely

• sos, and
• uvw.

As far as I can tell those two are methods for proving, for example, olympiad style inequalities.

I have mild misgivings about these tags for they seem to very specific. On the other hand:

• The tags may come in handy for someone preparing for a math-contest looking for training material,
• The probable creator (notified), to their credit, has written decent tag-wikis. This is a big plus in my eyes, because too often new tags have been created by a well-meaning but misguided badge hunter.

So I guess I am undecided whether these are good tags or not. Posting this partly to draw the creator's attention to this.

Answer 3

I want to create a new tag EV-Method, but I want to know before, what Community thinks about it.
This method is very useful for the proof of hard symmetric inequalities with $n$ variables. I think if our user will want to learn this method he'll can click this tag and see many examples, how to use this method. There is a big problem with examples and by creating of this tag we'll solve this problem.

Without this tag I think it's impossible to find these examples in the tag "inequality".

Answer 4

I think these tags (uvw and sos) are useful for the forum. User, which looks for to learn these methods, can click these tags and see many examples, how he can prove inequalities by these methods.

For example.

Let we need to prove that $$a\sqrt{a^2+bc}+b\sqrt{b^2+ac}+c\sqrt{c^2+ab}\geq\sqrt{2(a^2+b^2+c^2)(ab+ac+bc)}$$ for non-negatives $a$, $b$ and $c$.

A proof by SOS:

We need to prove that:

$$\sum\limits_{cyc}a\sqrt{a^2+bc}\geq\sqrt{2(a^2+b^2+c^2)(ab+ac+bc)}$$ or

$$\sum\limits_{cyc}\left(a^4+a^2bc+2ab\sqrt{(a^2+bc)(b^2+ac)}\right)\geq\sum\limits_{cyc}(2a^3b+2a^3c+2a^2bc)$$ or

$$\sum\limits_{cyc}(a^4-a^3b-a^3c+a^2bc)\geq\sum\limits_{cyc}\left(a^3b+a^3c+2a^2bc-2ab\sqrt{(a^2+bc)(b^2+ac)}\right)$$ or

$$\frac{1}{2}\sum\limits_{cyc}(a-b)^2(a+b-c)^2\geq\sum\limits_{cyc}ab\left(a^2+bc+b^2+ac-2\sqrt{(a^2+bc)(b^2+ac)}\right)$$ or

$$\sum\limits_{cyc}(a-b)^2(a+b-c)^2\geq2\sum\limits_{cyc}ab\left(\sqrt{a^2+bc}-\sqrt{b^2+ac}\right)^2$$ or

$$\sum\limits_{cyc}(a-b)^2(a+b-c)^2\left(1-\frac{2ab}{\left(\sqrt{a^2+bc}+\sqrt{b^2+ac}\right)^2}\right)\geq0,$$ which is obvious.

Now, try to find another examples for using SOS in the forum. For which it's enough to click the tag SOS and you'll see a full picture. I think it's impossible to see this picture without tag SOS.

Since the uvw method is much more useful than SOS (I think that particularly because of this method, inequalities were removed from IMO in the last six years), this tag also would be very useful for the forum.