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I came across this interesting inequality, and was looking for interesting proofs. $x,y,z \geq 0$

$$ 2\sqrt{\frac{x^{2}+y^{2}+z^{2}}{3}}+3\sqrt [3]{xyz}\leq 5\left(\frac{x+y+z}{3}\right) $$

Addendum.

In general, when is

$$ a\sqrt{\frac{x^{2}+y^{2}+z^{2}}{3}}+b\sqrt [3]{xyz}\leq (a+b)\left(\frac{x+y+z}{3}\right) $$

true?

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    I'm not sure what is interesting here, nor how to measure the interest found in a proof.2011-07-26
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    I deleted my answer because it was thoroughly invalid.2011-07-26
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    @Asaf, These inequalities are usually abbreviated as $2QM+3GM \leq 5AM$ The interesting bit is that it is true that $GM \leq AM \leq QM$.2011-07-26
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    @Asaf: what's interesting is that if the denominator inside the square root is $3^2 = 9$, the inequality is trivially true by splitting into two parts (hence $2+3 = 5$). As stated it in not entirely obvious why the inequality should be true.2011-07-26
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    @Willie Wong: So there is a typo? The denominator should be $9$ right?2011-07-26
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    @Damien: There is no typo.2011-07-26
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    @picakhu: But squaring we want to show that $$\frac{4(x^2+y^2+z^2)}{3} \leq \frac{4}{9}(x+y+z)^{2}$$ which is false for $x = 1, y = 2, z =3$.2011-07-26
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    @Damien, compare what you wrote to picakhu's first comment.2011-07-26
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    @picakhu: Really interesting inequallity. Did you found a "not interesting" proof? (like expanding all terms and using Muirhead or Schur technically).2011-07-26
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    @Jineon, There is a proof, not as terrible as a muirhead, but still one that I can never see myself figuring out. I am looking for ways in which someone can intuitively prove this. http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=195938&p=1077973#p10779732011-07-26
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    @picakhu: Can you change your title both to something that better reflects the question and doesn't use 'interesting?' It seems obvious to me that you find the question interesting, or you wouldn't ask it.2011-07-26
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    @picakhu: What is so non-intuitive about the proofs there? Honey_S's proof uses mixing variables/smoothing, which is a standard approach for this kind of inequality; or even better, can_hang's approach by EV theorem. I'm not sure how others usually find best constants - but it seems fair that calculus must involve somehow since those constants are usually ugly. (aka non-integers in this case) Then using EV theorem would be a very natural approach.2011-07-26
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    @Soarer, can you elaborate, and perhaps post a solution. I have never used or seen EV (whatever that is).2011-07-26
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    I see. I guess you are not familiar with mixing variables/EV then. I'll post a comment on this shortly.2011-07-26
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    I think you can ask even a more general question. Denote by $(x_1,\ldots,x_n)_p$ the $p$-mean defined as $\left(\frac{1}{n}\sum x_i^p\right)^{1/p}$ with $p = 0$ the geometric mean and $p = \infty$ just $\max$, one can ask for which $c = c(p,q,r;n)$ we have $$c((x))_p+(1-c)((x))_q \leq ((x))_r$$ for fixed $p,q,r$. Clearly if $r \geq p,q$ the answer is for every $c \in [0,1]$, and for $r < p,q$ the answer is for no $c\in [0,1]$. The question is thus interesting when $r$ is between $p$ and $q$.2011-07-26
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    @Willie Wong: This is best dealt with by the (n-1)-EV theorem.2011-07-26
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    Intuitively, I am guessing $QM - AM \leq AM - GM$ (i.e $AM$ is closer to $QM$ than $GM$) and that would imply what you have. This is true for two variables I believe. For three, a proof approach: we can assume $xyz = 1$ and reduce it to an inequality in two variables. Of course, I haven't tried it myself (hence a comment), but seems promising.2011-07-26
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    @Aryabhata, what you are saying is that $2GM+2QM \leq 4AM$, and we know that $GM \leq AM$, adding these two gives the question. However I think your inequality is wrong, but if it was right, the question is trivial (from your ineq).2011-07-26
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    @picakhu: Thanks for the title change!2011-07-26
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    @picakhu: Yes, that is what I was thinking. It is definitely true for 2 variables, not so sure about 3 (hence the comment).2011-07-26

2 Answers 2

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This is not a direct answer to the question, but it's probably too long for a comment, so I'm leaving it as an answer. (In the comments it seems that OP was not familiar with the technique of mixing variables/smoothing, which was used by Honey_S in the link provided to solve the problem; or the (n-1)-EV theorem, so this answer would be a quick exposition of what they are.)

Mixing variables/smoothing:

In inequalities such as $f(a,b,c) \ge 0$, we seek to prove an inequality of the type $f(a,b,c) \ge f(t,t,c)$. We expect to iterate this inequality so that we can conclude the minimum would be attained when many variables are equal.

Example 1: (AM-GM inequality)

We want to show that if $a,b,c > 0$, then $a+b+c \ge 3(abc)^{1/3}$.

Proof. Consider

$f(a,b,c) = a + b + c - 3(abc)^{1/3}$

Now by 2-variable AM-GM, we see that $f(a,b,c) \ge f(\sqrt{ab}, \sqrt{ab}, c)$. You can then imagine that if we keep on doing such smoothing - i.e. next time replace $(\sqrt{ab}, c)$ with 2-tuple of their geometric mean for example, then in infinite time we reach the case where all three variables are equal, and that $f(a,b,c)$ attains its minimum when $a=b=c$, which is 0.

In general there are many choices of $t$. If there's an initial condition on $a+b+c$, you may want to change $(a,b)$ to $\left(\frac{a+b}{2}, \frac{a+b}{2} \right)$ or sometimes $(0,a+b)$ if you guess that equality case of the inequality involves 0. If there is an initial condition on $a^2+b^2+c^2$, you may change $(a,b)$ to $\left(\sqrt{\frac{a^2+b^2}{2}}, \sqrt{\frac{a^2+b^2}{2}}\right)$ etc.

In the AM-GM example, life is nice because $f(a,b,c) \ge f(t,t,c)$ holds unconditionally. Very often, this is not the case. For example, in Honey_S's solution in the link, after assuming $abc=1$ by homogenity he proved that

$f(a,b,c) \ge f(\sqrt{ab},\sqrt{ab},c)$

for

$ f\left({a,b,c}\right) = 5\left({a+b+c}\right)-2\sqrt{3\left({a^{2}+b^{2}+c^{2}}\right)} $

only when $c = \max (a,b,c)$. However in this case, we are left to show that $f(t,t,c) \ge 0$ under the condition $t^2c = 1$. This is a one-variable inequality easily handled by calculus.

If you want to see more examples of smoothing in action, check this thread and the four links in that post, this and this for example.

(n-1)-EV theorem:

This is a theorem that kills many olympiad inequalities. You can see Vasile Cirtoaje's original article here. Basically what it does is that after some tedious calculus checking, many inequalities actually attain its extremum when (n-1) of the n variables involved in the inequality are equal.(And we can use calculus to check the remaining case) Check out theorem 3 and its corollaries in the link.

If you want to see its applications, see the "Applications" section of Vasc's paper, and if you want some more, see here or maybe a recent post on this forum. I hope this would be enough for you now :)

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    Thanks for the two methods. This will be a fun read for me. :) Maybe I can teach some to my friends who are still young enough to tackle olympiads!2011-07-26
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Let $x+y+z=3u$, $xy+xz+yz=3v^2$ and $xyz=w^3$.

Hence, we need to prove that $f(w^3)\geq0$, where $f(w^3)=5u-3w-2\sqrt{3u^2-2v^2}$.

We see that $f$ is a decreasing function,

which says that it's enough to prove our inequality for a maximal value of $w^3$,

which happens for equality case of two variables.

Indeed, $x$, $y$ and $z$ are non-negative roots of the equation $$(X-x)(X-y)(X-z)=0$$ or $$X^3-3uX^2+3v^2X-w^3=0$$ or $$w^3=X^3-3uX^2+3v^2X$$ and we see that $w^3$ gets a maximal value,

when a line $Y=w^3$ is a tangent line to the graph of $Y=X^3-3uX^2+3v^2X$,

which happens for equality case of two variables.

Since our inequality is homogeneous, we can assume $x=y=1$ and $z=t^3$.

Thus, it's enough to prove that: $$\frac{5}{3}(t^3+2)-3t\geq2\sqrt{\frac{t^6+2}{3}}$$ or $$(5t^3-9t+10)^2\geq12(t^6+2)$$ or $$(t-1)^2(13t^4+26t^3-51t^2-28t+76)\geq0,$$ which is obvious.

Done!

By the same way we can get a best values of $a$ and $b$.