The UVW-method

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1312 posts

#1 h May 26, 2009, 7:14 PM • 82 Y

Y by AwesomeToad, antimonyarsenide, airplanes1, Amir Hossein, nhatquangsin, truonghung, fractals, Alexey, dantx5, huiget, shinny98NT, utsab001, Saptashwa, BMSAUT, IMO2014, Samurott, Jettywang828, Dinesh, CaptainFlint, TheMaskedMagician, jh235, hamup1, hwl0304, 62861, Magikarp1, DrMath, Iliopoulos, hesa57, rjiang16, StarFrost7, mssmath, tarzanjunior, daisyxixi, CeuAzul, ineX, acegikmoqsuwy2000, memc38123, Sumgato, Willie, joey8189681, YadisBeles, Cube1, arandomperson123, ElevenCubed, VadimM, arulxz, Supercali, bel.jad5, Hexagrammum16, mathleticguyyy, Imayormaynotknowcalculus, OlympusHero, fjm30, Eliot, itslumi, Math_fj, Jc426, Dragunov, qwertyboyfromalotoftime, Icenight, Lamboreghini, sabkx, IraeVid13, Adventure10, Mango247, kiyoras_2001, Sedro, ISIS121, and 14 other users

Here it is.

Ten pages about the uvw-method. First introducing the same method as arqady has used, but in this is also included another method, which really shortens solutions and make them "nicer". I introduce a new and easier way to prove some inequalities, using that fixing $a+b+c$ and $ab+bc+ca$ we know when $abc$ assumes its maximum and minimum! This can give us some terribly short and easy proofs!

Enjoy!

Btw. Please post any corrections you may find!

Attachments:

The uvw method.pdf (134kb)

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Dr Sonnhard Graubner

16100 posts

Dr Sonnhard Graubner

#2 h May 26, 2009, 7:19 PM • 27 Y

Y by antimonyarsenide, dantx5, va2010, CatalinBordea, hwl0304, efang, DrMath, rjiang16, mikechen, bestwillcui1, acegikmoqsuwy2000, niraekjs, chocopuff, mathleticguyyy, Imayormaynotknowcalculus, OlympusHero, Eliot, Lamboreghini, Adventure10, Mango247, and 7 other users

hello Mathias, thank you very much for your article, i think, it is nice, i like the uvw method.
Sonnhard.

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Dimitris X

599 posts

Dimitris X

#3 h May 26, 2009, 7:49 PM • 3 Y

Y by dantx5, Akatsuki1010, Adventure10

Thank you very much too Mathias.....
I would like to learn more about this method

Dimitris

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puuhikki

979 posts

puuhikki

#4 h May 26, 2009, 8:06 PM • 11 Y

Y by dantx5, Wizard_32, Vasu090, IraeVid13, Adventure10, Mango247, and 5 other users

I have no motivation to read the whole article. Here are some things that came into my mind when skimming.

I think the article would be more readable if you would put some long equations to their own lines and align some lines differently. In the page 6 you define $f(t)$ twice! In page 4, It's not good to write text inside the brackets. Never start a sentence by mathematical symbol. Fix typos. In mathematical texts, one should try to avoid the use of implication or equivalence arrows. Vieta's, not vietés. Instead of writing "Will post full proof later.", please give an exact location. The equation at the end of page two continues on the next page.

It takes time to write a good mathematical paper.

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aauqen

17 posts

aauqen

#5 h May 27, 2009, 7:40 PM • 2 Y

Y by Adventure10, Mango247

puuhikki wrote:

I have no motivation to read the whole article.
...
It takes time to write a good mathematical paper.

Then don't. I would rather see it as early as possible than wait for some cosmetic improvements. I also see no violation in the article being posted; if Mathias_DK decides to clean it up, this early display of the article, possibly even attracting constructive critique, can do no harm.

Thank you, Mathias_DK, for your work. The only thing I didn't like is the convention to use u, v, w for these means - it's very easy to mix these numbers up in writing. Why not use i, j, k, for example? I understand that p, q, r are already taken for inequalities. Are there any other conventions I should know of?

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FelixD

588 posts

FelixD

#6 h May 27, 2009, 8:10 PM • 2 Y

Y by arandomperson123, Adventure10

Mathias chose $u$ , $v$ , $w$ because arqady, who discovered this method (on mathlinks, maybe someone else has discovered this method too, I don't know^^) introduced the letters $u$ , $v$ and $w$ .
And I completely agree, for handwriting $u$ , $v$ and $w$ were not the best chose^^, but why not use other letters at home?

Besides, in my opinion, $i$ and $j$ are even worse

every time I work with some indices, I use $i$ and $j$ and normally I get some troubles then

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puuhikki

979 posts

puuhikki

#7 h May 27, 2009, 8:12 PM • 1 Y

Y by Adventure10

aauqen wrote:

I also see no violation in the article being posted

Me neither, but Mathias_DK wanted to see all mistakes in the text. I just gave some of them. As he wants to write an article with no mistakes, he needs to improve the paper a lot.

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Bugi

1857 posts

Bugi

#8 h May 27, 2009, 8:20 PM • 2 Y

Y by Adventure10, Mango247

The article is great!

Well, if $u,v,w$ are the problem, don't use them. The point was not in letters, but in the idea of solving problems...

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FelixD

588 posts

FelixD

#9 h May 27, 2009, 8:29 PM • 3 Y

Y by Adventure10, Mango247, Mango247

If everybody reads this article, then it will be completely useless to post any symmetric inequality

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Mathias_DK

1312 posts

Mathias_DK

#10 h May 28, 2009, 8:54 AM • 2 Y

Y by Adventure10, Mango247

Hey Guys

Thank you for your comments! I will not edit the article right now, because I'm lazy

I published this article both to share it with the world, and to have something official, so that I could use it at IMO. Do you think that it is enough that the proof is published here, or should something else be done in order to have permission to use the ideas?

Felix: Fortunately not

The method can't solve the easy inequality $\frac{a^k+b^k+c^k}{3} \ge \left ( \frac{a+b+c}{3} \right ) ^k$ ...

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Dimitris X

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Dimitris X

#11 h May 28, 2009, 8:58 AM • 2 Y

Y by Adventure10, Mango247

Yes i have started reading it and it is really good...
But i would like to know who invented this powerfull method???
Arqady or someone else

:

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Mathias_DK

1312 posts

Mathias_DK

#12 h May 28, 2009, 9:13 AM • 1 Y

Y by Adventure10

Dimitris X wrote:

Yes i have started reading it and it is really good...
But i would like to know who invented this powerfull method???
Arqady or someone else

:

Arqady made the notations $3u=a+b+c,3v^2=ab+bc+ca,w^3=abc$ (Which I absolutely love

)
He also figured out that $3uv^2-2u^3-2\sqrt{(u^2-v^2)^3} \le w^3 \le 3uv^2-2u^3+2\sqrt{(u^2-v^2)^3}$ (Don't know if he was the first, but I have seen can_hang post something similar with the $p,q,r$ notation before.)
In my my paper I proved some things that is new to me (Of course I can't promise that someone else haven't found out before):
That there exists $a,b,c$ corresponding to $u,v,w$ if and only if $u^2 \ge v^2$ , $w^3 \in [3uv^2-2u^3-2\sqrt{(u^2-v^2)^3};3uv^2-2u^3+2\sqrt{(u^2-v^2)^3}]$
And the fact that we know that all of $u,v^2,w^3$ has a maximum and minimum value that can be archeived, plus that it is only archeived when two of $a,b,c$ are equal or when one of them are zero.

It was at first meant only as a guide on how to use the method, the way arqady did, but after I found the last thing I mentioned, I was more interested in that. (Mainly because it can give so short and easy to think of proofs. I don't use my computer when I do inequalities, so it's hard for me to calculate with squareroots as Arqady do with the methid

)

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MathHansung

4 posts

MathHansung

#13 h Jun 11, 2009, 12:32 AM • 1 Y

Y by Adventure10

While reading this text, I find a simple faults..

On page number2 , in the proof of Lemma1

If a=c=z (a complex number), then (a-b)(b-c)(c-a)=0 is a real number

Um... is it not a serious matter?

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Mathias_DK

1312 posts

Mathias_DK

#14 h Jun 11, 2009, 4:35 PM • 2 Y

Y by Adventure10, Mango247

MathHansung wrote:

While reading this text, I find a simple faults..

On page number2 , in the proof of Lemma1

If a=c=z (a complex number), then (a-b)(b-c)(c-a)=0 is a real number

Um... is it not a serious matter?

If $a=c=z$ then quite obviously $z \in \mathbb{R}$ . Could you elaborate?

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MathHansung

4 posts

MathHansung

#15 h Jun 12, 2009, 11:48 AM • 2 Y

Y by Adventure10, Mango247

in the lemma 1 of the proof of the uvw theorem

To prove that
" a, b, c are all real numbers " is equivalent to " (a-b)(b-c)(c-a) is a real number "

You first showed that
" If a,b,c are all real numbers , then so is (a-b)(b-c)(c-a) " ( of course it is trivial, as you mentioned )

then you tried to show that

" If one of a,b,c is not a real number , then (a-b)(b-c)(c-a) is also not a real number " ( actually you wrote that " I 'll show if a,b,c are not real numbers , then (a-b)(b-c)(c-a) is also not real number " but.. it's unclear sentence.. right?"
which is the contraposition of
" If (a-b)(b-c)(c-a) is a real number , then a,b,c are all real numbers."

However, there is a counterexample for
" If one of a,b,c is not a real number , then (a-b)(b-c)(c-a) is also not a real number "

in case of a=b=z ( z is an ideal number , - not a real number ), (a-b)(b-c)(c-a)=0 is a real number although on of a,b,c is not a real number.

you answered that
' If a=c=z then quite obviously z is a real number. '
but.. why?? ;

- I'm sorry to wirte all things with only words , without any symbols .. I don't know how to post some mathematical symbols;; And..
- My English can be too poor to understand.

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coolrg

648 posts

coolrg

#40 h Aug 7, 2015, 2:17 PM • 2 Y

Y by Adventure10, Mango247

Mathskidd wrote:

I am a beginner of learning uvw method.
When I read the last paragraph of page 6, section 4 application, I am stuck in the cubic equation
$x^3-6x^2-6x+28$ , which can get the roots $a,b,c$ .
Anyone may tell me how to search/ create it.
Thanks .

You can use the Rational Root Theorem to find that $x=2$ is a root of the cubic. Then, just divide the cubic by $x-2$ and use the Quadratic Formula to find the other two roots.

edit: oh oops, not what he/she was asking for

This post has been edited 1 time. Last edited by coolrg, Aug 7, 2015, 4:50 PM

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PKMathew

525 posts

PKMathew

#41 h Aug 7, 2015, 3:05 PM • 2 Y

Y by coolrg, Adventure10

@coolrg
I think mathskidd wanted to know how to find the cubic equation , instead of solving it.

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arqady

30150 posts

arqady

#42 h Nov 16, 2015, 1:06 PM • 2 Y

Y by adityaguharoy, Adventure10

The first using of $uvw$ in our forum:
http://artofproblemsolving.com/community/c6h44430p290751

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eshan

471 posts

eshan

#45 h Dec 13, 2015, 3:45 AM • 5 Y

Y by ineX, aops1221math, Amir Hossein, Adventure10, Mango247

I was studying the article today, and found the following mistake.

Does the inequality in the image really hold?

It does not hold for $a=b=c=3.$

Maybe you meant -

Unknown Origin. When $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1,~a,b,c\geqslant 0$ prove:
$\ (a-1)(b-1)(c-1)\geqslant 8.$ If this version is correct, then I have a solution.

My Solution.

The inequality is equivalent to $(1-a)(1-b)(1-c)\leqslant -8.$

Let us multiply both sides of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1,~a,b,c\geqslant 0$ by $abc.$ We get $ab+bc+ca=abc.$

Using the notations $a+b+c=3u,ab+bc+ca=3v^2,abc=w^3,$ we have $3v^2=w^3...(a)$

And, we know that

$(1-a)(1-b)(1-c)=(1-a-b+ab)(1-c)=1-(a+b+c)+(ab+bc+ca)-abc=1-3u+3v^2-w^3.$

Now using $(a)$ we get $(1-a)(1-b)(1-c)=1-3u.$

So we have $(1-a)(1-b)(1-c)\leqslant -8\iff 1-3u\leqslant -8\iff 3u\geqslant 9\iff a+b+c\geqslant 9.$

The last inequality is indeed true because it follows from $AM-GM$ that $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geqslant 9\iff a+b+c\geqslant 9.$
Now the proof is complete. $\Square$

Attachments:

This post has been edited 5 times. Last edited by eshan, Jan 14, 2016, 12:23 PM

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Akatsuki1010

1452 posts

Akatsuki1010

#46 h Aug 4, 2016, 2:02 PM • 1 Y

Y by Adventure10

Hi all, I have learned $uvw$ method for 2 weeks! After reading Mathias's artical, I could prove some hard inequalities! But I don't know when can we use $uvw$ ? It can be used in every synmetric inequalities or some kind of inequalities?
Thanks for yours help!

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arqady

30150 posts

arqady

#47 h Aug 4, 2016, 2:09 PM • 1 Y

Y by Adventure10

You can try use $uvw$ for all symmetric inequality of three variables, but it not always works.

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Akatsuki1010

1452 posts

Akatsuki1010

#48 h Aug 4, 2016, 2:26 PM • 2 Y

Y by Adventure10, Mango247

arqady wrote:

You can try use $uvw$ for all symmetric inequality of three variables, but it not always works.

Oh! I used it to prove your generalization inequalties but you asked me that what happened in that case and meant $uvw$ method didn't work with your inequalities, didn't it?

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Akatsuki1010

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Akatsuki1010

#49 h Aug 4, 2016, 2:27 PM • 2 Y

Y by Adventure10, Mango247

And it will not work in which case?

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Akatsuki1010

1452 posts

Akatsuki1010

#50 h Aug 8, 2016, 2:11 PM • 2 Y

Y by Adventure10, Mango247

Hello everyone, after prove the function of $3u$ , $3v^2$ , $w^3$ is convex or concave, in some case we have to check that $c$ or $b$ or $a\rightarrow0^+$ . Can anyone tell me which case we have to check that $c\rightarrow0^+$ ? Thank you!

This post has been edited 1 time. Last edited by Akatsuki1010, Aug 8, 2016, 2:11 PM

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ThuongPCT-A2

70 posts

ThuongPCT-A2

#51 h Aug 24, 2016, 1:29 PM • 1 Y

Y by Adventure10

arqady wrote:

You can try use $uvw$ for all symmetric inequality of three variables, but it not always works.

I think the UVW method is like ABC method
And it is only useful with the problem of symmetric =))

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Akatsuki1010

1452 posts

Akatsuki1010

#52 h Aug 29, 2016, 3:46 PM • 1 Y

Y by Adventure10

I think that we can use SOS Method after using $uvw$ theorem! It's very useful, right?

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Hermitianism

458 posts

Hermitianism

#53 h Mar 4, 2017, 8:46 AM • 2 Y

Y by Adventure10, Mango247

Akatsuki1010 wrote:

Hello everyone, after prove the function of $3u$ , $3v^2$ , $w^3$ is convex or concave, in some case we have to check that $c$ or $b$ or $a\rightarrow0^+$ . Can anyone tell me which case we have to check that $c\rightarrow0^+$ ? Thank you!

If $a,b,c$ are real ,you needn't check that.But if non-negative or positive,it behoves you to do.

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FISHMJ25

293 posts

FISHMJ25

#54 h Feb 16, 2019, 10:32 PM • 1 Y

Y by Adventure10

Very nice.

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ngv

251 posts

ngv

#55 h Apr 23, 2021, 8:53 PM • 1 Y

Y by Mango247

Mathias_DK wrote:

Here it is.

Ten pages about the uvw-method. First introducing the same method as arqady has used, but in this is also included another method, which really shortens solutions and make them "nicer". I introduce a new and easier way to prove some inequalities, using that fixing $a+b+c$ and $ab+bc+ca$ we know when $abc$ assumes its maximum and minimum! This can give us some terribly short and easy proofs!

Enjoy!

Btw. Please post any corrections you may find!

Hey Tejs, I am writing a book on inequality and I wanted to use the uvw method and the theorems/problems if necessary from your document. Would you be ok with that? I will mention your article of course

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KhuongTrang

704 posts

KhuongTrang

#56 h May 30, 2023, 8:13 AM

Y by

Bump for a very interesting method. :love:

This post has been edited 1 time. Last edited by KhuongTrang, May 30, 2023, 8:14 AM

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