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

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 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 tag needs to become more organized.

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 was created in January and 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, 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 in the tag management thread. But since some other similar tags started appearing, it is probably better to discuss them in general.

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

EDIT2: And the tags , , , have been created a few days later.

EDIT3: The tags and 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.)

• The young-inequality tag is very young you say?
– Asaf Karagila Mod
Apr 14, 2017 at 12:44
• @Asaf otherwise it would be holder-ineaquality, wouldn't it?
– quid Mod
Apr 14, 2017 at 16:40
• And the downside of creating a new tag is...? Apr 19, 2017 at 21:49
• – Asaf Karagila Mod
Apr 22, 2017 at 17:04
• @Mehrdad The answer Asaf has linked to addresses your question. And I have also tried to write something about this here. Apr 23, 2017 at 6:21
• Holy cow, that's a long answer to such a short question. =P I wasn't trying to ask what the downside is in general; I was trying to ask what the downside is for the inequalities you mention. "Inequalities" is hardly appropriate for something about AM-GM, Cauchy-Schwartz, or Jensen's inequality, and something like "metric spaces" or "convex analysis" would be pretty overkill (though perhaps not entirely inappropriate). My view is that they're all used commonly enough that they would pretty obviously flourish on their own. Apr 23, 2017 at 7:23
• @Mehrdad So I misunderstood your question. But the general answer also answer the specific question. Too many too specific tags cause problems. So it is good to discuss which inequalities should have their own tag, so that we do not end up with many tags which are not really used. (BTW I think that (inequalities) tag is definitely the most suitable for the questions about AM-GM etc. The question is whether to create additional tags.) Apr 23, 2017 at 7:28
• @Mehrdad Perhaps we could say that the question in your comment is very similar to what I am asking in the question. Anyway, if you think that for example the three equations you mention deserve to have a separate tag, you can post that as an answer - this is exactly what I am asking and from reaction of other users to your answer we would see whether they agree with you.) Also, if you wish, we can continue this discussion (general or specific to inequalities) in chat. Apr 23, 2017 at 7:31
• @Martin Sleziak Why you did not ask me about these things? Why are you doing this behind my back? I think it's not fair. Aug 7, 2017 at 10:56
• @MichaelRozenberg I do not think that what you are saying is accurate. The whole discussion started by the tags (a.m.-g.m.-inequality) created by Harsh Kumar, and (young-inequality) created by Leila. (I believe I notified Harsh Kumar about my post in tag management thread. So the tag creator was aware of this.) The post was publicly visible here on meta, and it appeared even in community bulletin. Anyway, if needed, we can discuss this in chat. Aug 7, 2017 at 12:11

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.

• What exactly would be tagged with say triangle-inequality and who should care to follow that tag specifically to see questions like: "Is there a triangle with side-length 3, 11, 29? I believe I need to use the triangle inequality." and "How can I show that the ring of p-adic integers is an ultra-metric space? I know that in an ultrametric space a stronger form of the triangle inequality needs to hold." side by side?
– quid Mod
Apr 23, 2017 at 9:51
• @quid: Same kind of people who presumably already care to follow "convex analysis" and see questions like "Proof that exponential function is convex" and "Legendre Transformation of a Lagrangian in Classical Mechanics" side-by-side? Apr 23, 2017 at 9:58
• While this is somewhat similar. on the one hand the situation for triangle-inequality would be more extreme, and on the other hand, I for one would in fact argue that it'd be better if the first type of question would not be tagged convex-analysis. (Likely it also never was actively tagged convex-analysis; alas there are synonyms, which I do not find all that useful, and the wiki somewhat allows that use.) Note the first question initially also was (mis)tagged set-theory, and certainly those that follow set-theory do not follow it for this type of question.
– quid Mod
Apr 23, 2017 at 10:11
• To put this differently while there obviously are people that are experts in convex analysis, I have some difficulty to imagine what an expert on the the triangle inequality should be.
– quid Mod
Apr 23, 2017 at 10:14
• @quid: I would expect there to exist experts on triangle inequality about as much as there would on Cauchy-Schwartz, and you seem to be okay with the latter, so... Apr 23, 2017 at 10:23
• I am not okay with C-S either. I just focused on one example you gave, admittedly the one that I find the worst. By the way there is no T there, just Schwarz.
– quid Mod
Apr 23, 2017 at 10:31
• @quid: Whoops, I used to spell it right but sometime in the past few months I started spelling it wrong again. Well, in that case, maybe you should first propose a common ground we can both stand on. Otherwise I'm just wasting my time rebutting against your arguments when you just keep rejecting everything and I have no idea what your stance on anything is. To me it's plain as day that Cauchy-Schwarz and Jensen deserve their own own tags, and the triangle inequality is a corollary... Apr 23, 2017 at 10:58
• @quid I know that several users consider this as one of the test whether a tag might be useful: "Can you imagine that somebody ads this tag among favorited or ignored tags?" (I was able to find one mention of this here.) Personally, I would consider a tag useful if it greatly enhances searching for questions, even in the case where nobody would follow it or when it is unclear who would be expert in given tag. (I am mentioning this since you seem to be using this test implicitly.) Apr 23, 2017 at 11:26
• As a side note, to me the possible (triangle-inequality) tag seems closer to general topology, metric spaces and normed spaces. So it is slightly outside of the originally suggested scope of the question, which was inequalities for real numbers. (Of course, it is true that Cauchy-Schwarz also make sense in general inner product spaces. But for C-S I would expect more questions about real numbers. If a tag triangle-inequality is created, my expectation is that it will mostly appear in questions related to metric and normed spaces.) Apr 23, 2017 at 11:28
• @MartinSleziak: I can totally imagine these being among the favorite or ignored tags. In fact, I would have probably put most of these metric-space inequalities in my own ignored tags if I had been regular enough of a user of Math.SE to have any favorite or ignored tags at all. On the other hand I could see myself putting something like the minimax (aka min-max) inequality in my favorite tags. And yes, I totally agree with you on searchability; that's my main criterion for a tag as well. And tagging something as C-S most definitely would help in that regard. Apr 23, 2017 at 12:19

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

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.

• Re: Posting this partly to draw the creator's attention to this. I've noticed both creation of uvw and also sos and buffalo-way. Of course, I am grateful for letting me know, but it is probably more important what community as a whole thinks about the issue of specific-inequality tags (and some related tags). Aug 6, 2017 at 19:46
• @MartinSleziak I am not as well informed about our criteria of when to create new tags as you are. A goal was to bring Michael Rozenberg here. To educate him about our ways of clearing tag creations in advance, and also to give him a chance to speak up. Sorry, if I implied that you would not have noticed. I know that you pay attention to tags. Aug 6, 2017 at 21:01
• I certainly take that as a compliment, although I am not sure whether anybody know criteria for creating new tags, since different users have different opinion on this. I agree with the point you made elsewhere that it is quite natural that subcommunities around certain tags naturally form on this site. So I think that voice of users who are active in the (inequalities) tag is important here. So I am grateful that you notified Michael Rozenberg of this discussion. Aug 7, 2017 at 14:42

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".

• Don't have any strong opinion on this (and btw I did not cast any votes in this thread). However, I think it might make sense to insert something related to inequality in the tag, like a.m.-g.m.-inequality or cauchy-schwarz-inequality have now. This would make it easier for both posters and later readers to find the right tag. Just sos or ev-method is a bit too cryptic for anyone who doesn't know already what they are looking for.
– dxiv
Aug 7, 2017 at 4:37
• To which extent can EV-method be considered standard (i.e., commonly used) name. For example, I do not see this name in Cvetkovski's book. Aug 8, 2017 at 5:34
• @Martin Sleziak In my opinion, this is a book for beginners. Aug 8, 2017 at 5:40
• @MichaelRozenberg Ok, nice put-down :-) Still the question remains - is the name EV-method standard? Is it used somewhere? Aug 8, 2017 at 5:44
• @Martin Sleziak We can not use this method in IMO without proof, but in IMC it's obligatory! This method found Vasile Cirtoje 10 years ago and it's very usefull. Aug 8, 2017 at 5:53
• To address @dxiv's objection, maybe equal-variable-method or equal-variable-theorem would be a bit more descriptive name, if the tag is going to be created. (After all, ev-method can be added as a synonym.) Aug 8, 2017 at 6:36
• @Martin Sleziak I think the first is better. Aug 8, 2017 at 6:37

I think these tags ( and ) 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.

• I haven't voted on this post, but I suspect that the downvotes are due to you not discussing the usefulness of uvw/sos as TAGS rather than as techniques of doing math. Aug 7, 2017 at 4:15
• Michael, my advice would be that before creating new tags you should read all of this. I am not the only one who would be willing to grant Martin Sleziak dictatorial powers re tagging. He knows what he's doing, and has earned community respect (not only) in this matter. Aug 7, 2017 at 7:01
• To add to Jyrki's nice comments: you might want to read this and this, and then think carefully about whether tagging a question based on the techniques of only one of the answers in it is sensible. Aug 7, 2017 at 7:44
• @J. M. isn't a mathematician I just read it now and I think it exactly which I am doing, when I am adding or I am creating a new tag. Can you explain me where do you see a problem? Aug 7, 2017 at 8:12
• @MichaelRozenberg I am grateful that you found time to join the discussion - it is good to hear from people active in the inequalities tag (perhaps also contest-math). And also thanks for editing this post. (So far I did not vote on any of the answers here, I'll take some time thinking about these new tags before doing so.) Aug 7, 2017 at 14:48
• I haven't said there was a problem, Michael (I haven't voted any of your answers either). I only linked to those as a reminder. If, as you say, you can fully justify these tags to everybody else, then I've no objections. Aug 7, 2017 at 14:57
• Re: 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. Certainly, there are many other methods how to create a collection of examples for some methods than using tags. Aug 8, 2017 at 7:30
• @Martin Sleziak What do you mean? Give me example. Aug 8, 2017 at 7:31
• For example, various lists of problems. With additional advantage that they can be better organized and comments can be included. They can be created outside of math.SE, sometimes they are tolerated also here. See, for example, this question and other posts linked there. Aug 8, 2017 at 7:37
• @Martin Sleziak I think it's very bad idea for two reasons. 1. Adding question in live it's the best thing! It much better than a book. This thing we have because there is tagging. 2. We can else to make these comments and links. I think it's a big and unnecessary work. Aug 8, 2017 at 7:40
• @MichaelRozenberg If you wish to discuss question collections in more detail, we can move this to chat - because we digressed from the original topic which is tagging. (Although I am not sure I have much to add to what is written in the posts I linked above.) Aug 8, 2017 at 9:33