82
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$x,y,z >0$, prove $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$

Note: Often Stack Exchange asked to show some work before answering the question. This inequality was used as a proposal problem for National TST of an Asian country a few years back. However, upon receiving the official solution, the committee decided to drop this problem immediately. They don't believe that any students can solve this problem in 3 hour time frame.

Update 1: In this forum, some bragged that BW is the only solution for this problem, which is completely wrong and stupid. This problem is listed as "coffin problems" in my country. The official solution is very elementary and elegant.

Update 2: Although there are some solutions (or partial solution) based on numerical method, I am more interested in the approach with "pencil and papers." I think the approach by Peter Scholze in here may help.

Update 3: Michael has tried to apply Peter Scholze's method but not found the solution yet.

Update 4: Symbolic expanding with computer is employed and verify the inequality. However, detail solution that not involved computer has not been found. Whoever can solve this inequality using high school math knowledge will be considered as the "King of Inequality".

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  • 7
    $\begingroup$ I have no idea about the official solution. I try this problem for the past 3 years but not yet success. Even with brute force, I still cannot solve it. This shows level of insanity this problem has. $\endgroup$ – HN_NH May 7 '16 at 16:37
  • 8
    $\begingroup$ A simple observation is that the inequality is homogeneous, so it suffices to prove the case $ x + y + z = 1 $. Wolfram helped me solve the mess of an equation system that arises out of Lagrange multipliers, so I am convinced that the inequality is true now. (I have no idea how to solve the system myself, so this doesn't really count as a solution.) $\endgroup$ – Starfall May 7 '16 at 23:01
  • 2
    $\begingroup$ I believe that $\sum_{\mathrm{cyc}}\frac{x^4}{ay^3+bz^3}\geq\frac{x+y+z}{a+b}$, and why not $\sum_{\mathrm{cyc}}\frac{x_n^4}{a_1x_1^3+\ldots a_{n-1}x_{n-1}^3}\geq\frac{x_1+\ldots+x_{n-1}+x_n}{a_1+\ldots+a_{n-1}+a_n}$ for $n\geq3$. Maybe the general case could be somehow helpful... $\endgroup$ – Nicolas May 22 '16 at 9:13
  • 1
    $\begingroup$ Any further ideas for changes of the tags here must be first proposed to me. $\endgroup$ – quid Oct 13 '18 at 23:22
  • 1
    $\begingroup$ @max8128 I solved this problem three years ago and posted it here. See my solution. $\endgroup$ – Michael Rozenberg Feb 18 at 21:49

12 Answers 12

16
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This inequality is very strong and I am sure that it's impossible to prove it during competition without calculator.

A big problem we get around $(x,y,z)=(0.822,1.265,1.855)$.

By the way, the Buffalo Way helps:

Let $x=\min\{x,y,z\}$, $y=x+u$,$z=x+v$ and $x=t\sqrt{uv}$.

Hence, $\frac{13}{5}\prod\limits_{cyc}(8x^3+5y^3)\left(\sum\limits_{cyc}\frac{x^4}{8x^3+5y^3}-\frac{x+y+z}{13}\right)=$

$$=156(u^2-uv+v^2)x^8+6(65u^3+189u^2v-176uv^2+65v^3)x^7+$$ $$+2(377u^4+1206u^3v+585u^2v^2-1349uv^3+377v^4)x^6+$$ $$+3(247u^5+999u^4v+1168u^3v^2-472u^2v^3-726uv^4+247)x^5+$$ $$+3(117u^6+696u^5v+1479u^4v^2+182u^3v^3-686u^2v^4-163uv^5+117v^6)x^4+$$ $$+(65u^7+768u^6v+2808u^5v^2+2079u^4v^3-1286u^3v^4-585u^2v^5+181uv^6+65v^7)x^3+$$$$+3uv(40u^6+296u^5v+472u^4v^2-225u^2v^4+55uv^5+25v^6)x^2+ $$ $$+u^2v^2(120u^5+376u^4v+240u^3v^2-240u^2v^3-25uv^4+75v^5)x+$$ $$+5u^3v^3(8u^4+8u^3v-8uv^3+5v^4)\geq$$ $$\geq u^5v^5(156t^8+531t^7+2t^6-632t^5-152t^4+867t^3+834t^2+299t+40)\geq0$$

Done!

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  • 1
    $\begingroup$ It seems that everybody overlooked that this is the solutions of this problem. I changed the last step of the proof so hat it is simpler. I hope thi is ok. I also added an answer that is an extenede comment to this proof the shows how to confirm these calculations with the cas Maxima. $\endgroup$ – miracle173 Jul 3 '16 at 12:16
  • $\begingroup$ there is an error in my edit. One cannot assume $z\le y$. I try do undo the change. $\endgroup$ – miracle173 Jul 3 '16 at 13:06
  • $\begingroup$ @MichaelRozenberg: can you elaborate how you get the the polynomial in t and the estimates? $\endgroup$ – miracle173 Jul 3 '16 at 13:22
  • $\begingroup$ @miracle173 for example, $156(u^2-uv+v^2)x^8\geq156uvx^8=156u^5v^5t^8$. $\endgroup$ – Michael Rozenberg Jul 3 '16 at 13:31
  • $\begingroup$ Here is a link to an answer that contains a Maxima script with some details to this answer. $\endgroup$ – miracle173 Jul 3 '16 at 18:30
15
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enter image description here

This is a question of the symmetric type, such as listed in:

With a constraint $\;x+y+z=1\;$ and $\;x,y,z > 0$ . Sort of a general method to transform such a constraint into the inside of a triangle in 2-D has been explained at length in:

Our function $f$ in this case is: $$ f(x,y,z) = \frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} - \frac{1}{13} $$ And the minimum of that function inside the abovementioned triangle must shown to be greater or equal to zero. Due to symmetry - why oh why can it not be proved with Group Theory - an absolute minimum of the function is expected at $(x,y,z) = (1/3,1/3,1/3)$. Another proof without words is attempted by plotting a contour map of the function, as depicted. Levels (nivo) of these isolines are defined (in Delphi Pascal) as:

nivo := min + sqr(g/grens)*(max-min); { sqr = square ; grens = 20 ; g = 0..grens }
The whiteness of the isolines is proportional to the (positive) function values; they are almost black near the minimum and almost white near the maximum values. Maximum and minimum values of the function are observed to be:

 0.00000000000000E+0000 < f < 4.80709198767699E-0002
The little $\color{blue}{\mbox{blue}}$ spot in the middle is where $\,0 \le f(x,y,z) < 0.00002$ .

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  • 1
    $\begingroup$ Interesting technique. Thank you very much for trying my inequality. One up vote !!! $\endgroup$ – HN_NH May 24 '16 at 6:05
13
+25
$\begingroup$

Too long for a comment.


The Engel form of Cauchy-Schwarz is not the right way:

$$\frac{(x^2)^2}{8x^3+5y^3}+\frac{(y^2)^2}{8y^3+5z^3}+\frac{(z^2)^2}{8z^3+5x^3} \geq \frac{(x^2+y^2+z^2)^2}{13(x^3+y^3+z^3)}$$

So we should prove that $$\frac{(x^2+y^2+z^2)^2}{13(x^3+y^3+z^3)}\geq\frac{x+y+z}{13}$$

which is equivalent to $$\frac{(x^2+y^2+z^2)^2}{(x^3+y^3+z^3)}\geq x+y+z$$ but by Cauchy-Schwarz again we have $$x+y+z=\frac{(x^2)^2}{x^3} +\frac{(y^2)^2}{y^3}+\frac{(z^2)^2}{z^3} \geq \frac{(x^2+y^2+z^2)^2}{(x^3+y^3+z^3)}$$

and the inequalities are in the wrong way.

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  • 2
    $\begingroup$ So directly using CS is too weak. Any other stronger inequalities that we can try? $\endgroup$ – Colescu May 22 '16 at 5:56
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    $\begingroup$ I've been asked to delete my answer, and I want to explain why I posted it. I know that this is not an answer to the original problem. I posted it because there where another anwer claiming that the problem could have been solved by directly applying the Cauchy Schwarz inequality. So I though that was a good idea to show that this way is not the right way. Now the other answer has been deleted, but I still think that it can be useful to note that a direct application of CS does not work. $\endgroup$ – user126154 Jun 23 '16 at 9:03
  • $\begingroup$ This is very interesting... instead of an answer showing what works, we have an answer showing what doesn't work, which could prove to be equally useful! Love it! $(+1)$ :D $\endgroup$ – Mr Pie Feb 2 at 13:58
5
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This is more an extended comment to the answer of @MichaelRozenberg than an answer by its own. I used a short Maxima to confirm the equation derived by @MichaelRozenberg. I used Maxima because it is open source.

Here is the Maxima script (statements are terminated by $ or by ;):


"I use string to comment this file"$

"the flag `display2d`  controls 
the display of the output. You can unset it (display2d:false), that makes it easy to copy 
the maxima output to math.stackexchange"$

"to make it easier to input the problem data 
we define to function g and f:"$

g(r,s):=(8*r^3+5*s^3);

f(r,s):=r^4/g(r,s);

"
the initial problem has the form 
L(x,y,t)>=R(x,y,z) 
but we subtract R(x,y,z) from this equation and 
we state the problem in the form 
term0>=0 
where term0 is L(x,y,z)-R(x,y,z) 
this is term0:
"$

term0:f(x,y)+f(y,z)+f(z,x)-(x+y+z)/13;

"
Now we multiply the term0 by a positive fraction of the (positive) common denominator 
and get term1 that satisfies 
term1>=0 
`ratsimp` does some simplification like cancelling 
"$

term1:13/5*g(x,y)*g(y,z)*g(z,x)*term0,ratsimp;

"
now we assume x=0 and v>=0
`,y=x+u` and `,z=x+v` do these substitutions
"$

term2:term1,y=x+u,z=x+v;

"
ratsimp(.,x) does some simplification and displays the term as polynomial of x
"$

term3:ratsimp(term2,x);

for p:0 thru hipow(term3,x) do print (coeff(term3,x,p)*x^p);

"the lowerbound polynomial is given by @Michael Rozenberg";

lowerbound:u^5*v^5*(156*t^8+531*t^7+2*t^6-632*t^5-152*t^4+867*t^3+834*t^2+299*t+40);

"we use the expanded version of the lowerbound polynomial";

lb:lowerbound,expand;

"we want to avoid squareroots and therefore substitute u bei `q^2` and v by `w^2`. 
The expression `sqrt(u*v)` (see thhe proof of Michael Rozenberg) then can be replaced by q*w";

"We want to avoid squareroots and therefore substitute u bei `q^2` and v by `w^2`. 
The expression `sqrt(u*v)` (see thhe proof of Michael Rozenberg) then can be replaced by q*w.
The following loop checks for each exponent k, that the coefficient of the original polynomial 
in x (adjusted by sqrt(u*v)^k) is larger than the coeffiecient of the lowerbound polynomial.
This value is called wdiff in the following.
We already mentioned that we do not use the original variable u and v but first transform 
to q and w as described above and therefor the adjustment is (q*w)^k  instead of sqrt(u*v)^k.
`wdiff` is a homogenous polynom of degree 20. We devide by `w`and replace `q/w` by `s`
and get the polynomial `poly` with vrailbe `s`. For these polynomials we calculate the number
of roots greater than 0. This can be done bei the `nroot` function that uses  'sturm's theorem' 
Then we  calculate the value of poly at 2. If this value is greate 0 and there are 
no zeros greater 0 then wdiff is greater or equal 0 for all nonnegative q and w and therefore 
for all nonegative u and v. This was what we wanted to proof.
We see that all polynomails are positive at 2 and also for all except for k=8 there are no zeros
greater than 0. For k=8 we have a zero with even multiplicity.
";

for k:0 thru 8 do (
    coff_x:coeff(term3,x,k),
    coeff_t:coeff(lb,t,k),
    wdiff:ev(coff_x*(q*w)^k-coeff_t,u=q^2,v=w^2),
    poly:ratsubst(s,q/w,expand(wdiff/w^20)),
    nr:nroots(poly,0,inf),
    print("==="),
    print("k=",k),
    print("coeff(term3, x,",k,")=",coff_x),
    print("coeff(lb, t,",k,")=",coeff_t),
    print("wdiff=",wdiff),
    print("polynomial:",poly),
    print("factors=",factor(poly)),
    print("number of roots >0:",nr),
    print("poly(2)=",ev(poly,s=2))
    );

"finally we proof that the lowerbbound polynomial has no positive root and that 
it is greater than 0 for t=1. Therefor it is greater or equal than 0 for all admissible values";

poly:ratcoeff(lowerbound,u^5*v^5);

poly,t=1;

nroots(poly,0,inf);




I ran the scrip on the Xmaxima console and get the following output. I use this console with this rather ugly kind of output because it can be simply copied and pasted to math.stackecchange. A prettier output can be found here at an online version of Maxima

(%i1) display2d:false;
(%o1) false
(%i2) 
read and interpret file: #pD:/maxima/ineq1775572.mac
(%i3) "I use string to comment this file"
(%i4) "the flag `display2d`  controls 

the display of the output. You can unset it (display2d:false), that makes it easy to copy 

the maxima output to math.stackexchange"
(%i5) "to make it easier to input the problem data 

we define to function g and f:"
(%i6) g(r,s):=8*r^3+5*s^3
(%o6) g(r,s):=8*r^3+5*s^3
(%i7) f(r,s):=r^4/g(r,s)
(%o7) f(r,s):=r^4/g(r,s)
(%i8) "

the initial problem has the form 

L(x,y,t)>=R(x,y,z) 

but we subtract R(x,y,z) from this equation and 

we state the problem in the form 

term0>=0 

where term0 is L(x,y,z)-R(x,y,z) 

this is term0:

"
(%i9) term0:f(x,y)+f(y,z)+f(z,x)+(-(x+y+z))/13
(%o9) z^4/(8*z^3+5*x^3)+y^4/(5*z^3+8*y^3)+((-z)-y-x)/13+x^4/(5*y^3+8*x^3)
(%i10) "

Now we multiply the term0 by a positive fraction of the (positive) common denominator 

and get term1 that satisfies 

term1>=0 

`ratsimp` does some simplification like cancelling 

"
(%i11) ev(term1:(13*g(x,y)*g(y,z)*g(z,x)*term0)/5,ratsimp)
(%o11) (25*y^3+40*x^3)*z^7+((-40*y^4)-40*x*y^3-64*x^3*y+40*x^4)*z^6
                          +(40*y^6+39*x^3*y^3-40*x^6)*z^4
                          +(40*y^7-64*x*y^6+39*x^3*y^4+39*x^4*y^3-40*x^6*y
                                  +25*x^7)
                           *z^3+((-40*x^3*y^6)-64*x^6*y^3)*z+25*x^3*y^7
                          -40*x^4*y^6+40*x^6*y^4+40*x^7*y^3
(%i12) "

now we assume x=0 and v>=0

`,y=x+u` and `,z=x+v` do these substitutions

"
(%i13) ev(term2:term1,y = x+u,z = x+v)
(%o13) (x+v)^3*(40*(x+u)^7-64*x*(x+u)^6+39*x^3*(x+u)^4+39*x^4*(x+u)^3+25*x^7
                          -40*x^6*(x+u))
 +25*x^3*(x+u)^7+(x+v)*((-40*x^3*(x+u)^6)-64*x^6*(x+u)^3)
 +(x+v)^4*(40*(x+u)^6+39*x^3*(x+u)^3-40*x^6)-40*x^4*(x+u)^6+40*x^6*(x+u)^4
 +(x+v)^6*((-40*(x+u)^4)-40*x*(x+u)^3+40*x^4-64*x^3*(x+u))
 +(x+v)^7*(25*(x+u)^3+40*x^3)+40*x^7*(x+u)^3
(%i14) "

ratsimp(.,x) does some simplification and displays the term as polynomial of x

"
(%i15) term3:ratsimp(term2,x)
(%o15) (156*v^2-156*u*v+156*u^2)*x^8+(390*v^3-1056*u*v^2+1134*u^2*v+390*u^3)
                                     *x^7
                                    +(754*v^4-2698*u*v^3+1170*u^2*v^2
                                             +2412*u^3*v+754*u^4)
                                     *x^6
                                    +(741*v^5-2178*u*v^4-1476*u^2*v^3
                                             +3504*u^3*v^2+2997*u^4*v+741*u^5)
                                     *x^5
                                    +(351*v^6-489*u*v^5-2058*u^2*v^4
                                             +546*u^3*v^3+4437*u^4*v^2
                                             +2088*u^5*v+351*u^6)
                                     *x^4
                                    +(65*v^7+181*u*v^6-585*u^2*v^5
                                            -1286*u^3*v^4+2079*u^4*v^3
                                            +2808*u^5*v^2+768*u^6*v+65*u^7)
                                     *x^3
                                    +(75*u*v^7+165*u^2*v^6-675*u^3*v^5
                                              +1416*u^5*v^3+888*u^6*v^2
                                              +120*u^7*v)
                                     *x^2
                                    +(75*u^2*v^7-25*u^3*v^6-240*u^4*v^5
                                                +240*u^5*v^4+376*u^6*v^3
                                                +120*u^7*v^2)
                                     *x+25*u^3*v^7-40*u^4*v^6+40*u^6*v^4
                                    +40*u^7*v^3
(%i16) for p from 0 thru hipow(term3,x) do print(coeff(term3,x,p)*x^p)
25*u^3*v^7-40*u^4*v^6+40*u^6*v^4+40*u^7*v^3 
(75*u^2*v^7-25*u^3*v^6-240*u^4*v^5+240*u^5*v^4+376*u^6*v^3+120*u^7*v^2)*x 
(75*u*v^7+165*u^2*v^6-675*u^3*v^5+1416*u^5*v^3+888*u^6*v^2+120*u^7*v)*x^2 
(65*v^7+181*u*v^6-585*u^2*v^5-1286*u^3*v^4+2079*u^4*v^3+2808*u^5*v^2+768*u^6*v
       +65*u^7)
 *x^3

(351*v^6-489*u*v^5-2058*u^2*v^4+546*u^3*v^3+4437*u^4*v^2+2088*u^5*v+351*u^6)
 *x^4

(741*v^5-2178*u*v^4-1476*u^2*v^3+3504*u^3*v^2+2997*u^4*v+741*u^5)*x^5 
(754*v^4-2698*u*v^3+1170*u^2*v^2+2412*u^3*v+754*u^4)*x^6 
(390*v^3-1056*u*v^2+1134*u^2*v+390*u^3)*x^7 
(156*v^2-156*u*v+156*u^2)*x^8 
(%o16) done
(%i17) "the lowerbound polynomial is given by @Michael Rozenberg"
(%o17) "the lowerbound polynomial is given by @Michael Rozenberg"
(%i18) lowerbound:u^5*v^5
                     *(156*t^8+531*t^7+2*t^6-632*t^5-152*t^4+867*t^3+834*t^2
                              +299*t+40)
(%o18) (156*t^8+531*t^7+2*t^6-632*t^5-152*t^4+867*t^3+834*t^2+299*t+40)*u^5*v
                                                                            ^5
(%i19) "we use the expanded version of the lowerbound polynomial"
(%o19) "we use the expanded version of the lowerbound polynomial"
(%i20) ev(lb:lowerbound,expand)
(%o20) 156*t^8*u^5*v^5+531*t^7*u^5*v^5+2*t^6*u^5*v^5-632*t^5*u^5*v^5
                      -152*t^4*u^5*v^5+867*t^3*u^5*v^5+834*t^2*u^5*v^5
                      +299*t*u^5*v^5+40*u^5*v^5
(%i21) "we want to avoid suareroots and therefore substitute u bei `q^2` and v by `w^2`. 

The expression `sqrt(u*v)` (see thhe proof of Michael Rozenberg) then can be replaced by q*w"
(%o21) "we want to avoid suareroots and therefore substitute u bei `q^2` and v by `w^2`. 

The expression `sqrt(u*v)` (see thhe proof of Michael Rozenberg) then can be replaced by q*w"
(%i22) "We want to avoid suareroots and therefore substitute u bei `q^2` and v by `w^2`. 

The expression `sqrt(u*v)` (see thhe proof of Michael Rozenberg) then can be replaced by q*w.

The following loop checks for each exponent k, that the coefficient of the original polynomial 

in x (adjusted by sqrt(u*v)^k) is larger than the coeffiecient of the lowerbound polynomial.

This value is called wdiff in the following.

We already mentioned that we do not use the original variable u and v but first transform 

to q and w as described above and therefor the adjustment is (q*w)^k  instead of sqrt(u*v)^k.

`wdiff` is a homogenous polynom of degree 20. We devide by `w`and replace `q/w` by `s`

and get the polynomial `poly` with vrailbe `s`. For these polynomials we calculate the number

of roots greater than 0. This can be done bei the `nroot` function that uses  'sturm's theorem' 

Then we  calculate the value of poly at 2. If this value is greate 0 and there are 

no zeros greater 0 then wdiff is greater or equal 0 for all nonnegative q and w and therefore 

for all nonegative u and v. This was what we wanted to proof.

We see that all polynomails are positive at 2 and also for all except for k=8 there are no zeros

greater than 0. For k=8 we have a zero with even multiplicity.

"
(%o22) "We want to avoid suareroots and therefore substitute u bei `q^2` and v by `w^2`. 

The expression `sqrt(u*v)` (see thhe proof of Michael Rozenberg) then can be replaced by q*w.

The following loop checks for each exponent k, that the coefficient of the original polynomial 

in x (adjusted by sqrt(u*v)^k) is larger than the coeffiecient of the lowerbound polynomial.

This value is called wdiff in the following.

We already mentioned that we do not use the original variable u and v but first transform 

to q and w as described above and therefor the adjustment is (q*w)^k  instead of sqrt(u*v)^k.

`wdiff` is a homogenous polynom of degree 20. We devide by `w`and replace `q/w` by `s`

and get the polynomial `poly` with vrailbe `s`. For these polynomials we calculate the number

of roots greater than 0. This can be done bei the `nroot` function that uses  'sturm's theorem' 

Then we  calculate the value of poly at 2. If this value is greate 0 and there are 

no zeros greater 0 then wdiff is greater or equal 0 for all nonnegative q and w and therefore 

for all nonegative u and v. This was what we wanted to proof.

We see that all polynomails are positive at 2 and also for all except for k=8 there are no zeros

greater than 0. For k=8 we have a zero with even multiplicity.

"
(%i23) for k from 0 thru 8 do
           (coff_x:coeff(term3,x,k),coeff_t:coeff(lb,t,k),
            wdiff:ev(coff_x*(q*w)^k-coeff_t,u = q^2,v = w^2),
            poly:ratsubst(s,q/w,expand(wdiff/w^20)),nr:nroots(poly,0,inf),
            print("==="),print("k=",k),print("coeff(term3, x,",k,")=",coff_x),
            print("coeff(lb, t,",k,")=",coeff_t),print("wdiff=",wdiff),
            print("polynomial:",poly),print("factors=",factor(poly)),
            print("number of roots >0:",nr),print("poly(2)=",ev(poly,s = 2)))
=== 
k= 0 
coeff(term3, x, 0 )= 25*u^3*v^7-40*u^4*v^6+40*u^6*v^4+40*u^7*v^3 
coeff(lb, t, 0 )= 40*u^5*v^5 
wdiff= 25*q^6*w^14-40*q^8*w^12-40*q^10*w^10+40*q^12*w^8+40*q^14*w^6 
polynomial: 40*s^14+40*s^12-40*s^10-40*s^8+25*s^6 
factors= 5*s^6*(8*s^8+8*s^6-8*s^4-8*s^2+5) 
number of roots >0: 0 
poly(2)= 769600 
=== 
k= 1 
coeff(term3, x, 1 )= 
               75*u^2*v^7-25*u^3*v^6-240*u^4*v^5+240*u^5*v^4+376*u^6*v^3
                         +120*u^7*v^2 
coeff(lb, t, 1 )= 299*u^5*v^5 
wdiff= 
      q*w
       *(75*q^4*w^14-25*q^6*w^12-240*q^8*w^10+240*q^10*w^8+376*q^12*w^6
                    +120*q^14*w^4)
       -299*q^10*w^10 
polynomial: 120*s^15+376*s^13+240*s^11-299*s^10-240*s^9-25*s^7+75*s^5 
factors= s^5*(120*s^10+376*s^8+240*s^6-299*s^5-240*s^4-25*s^2+75) 
number of roots >0: 0 
poly(2)= 7074016 
=== 
k= 2 
coeff(term3, x, 2 )= 
               75*u*v^7+165*u^2*v^6-675*u^3*v^5+1416*u^5*v^3+888*u^6*v^2
                       +120*u^7*v 
coeff(lb, t, 2 )= 834*u^5*v^5 
wdiff= 
      q^2*w^2
         *(75*q^2*w^14+165*q^4*w^12-675*q^6*w^10+1416*q^10*w^6+888*q^12*w^4
                      +120*q^14*w^2)
       -834*q^10*w^10 
polynomial: 120*s^16+888*s^14+1416*s^12-834*s^10-675*s^8+165*s^6+75*s^4 
factors= 3*s^4*(40*s^12+296*s^10+472*s^8-278*s^6-225*s^4+55*s^2+25) 
number of roots >0: 0 
poly(2)= 27198192 
=== 
k= 3 
coeff(term3, x, 3 )= 
               65*v^7+181*u*v^6-585*u^2*v^5-1286*u^3*v^4+2079*u^4*v^3
                     +2808*u^5*v^2+768*u^6*v+65*u^7 
coeff(lb, t, 3 )= 867*u^5*v^5 
wdiff= 
      q^3*w^3
         *(65*w^14+181*q^2*w^12-585*q^4*w^10-1286*q^6*w^8+2079*q^8*w^6
                  +2808*q^10*w^4+768*q^12*w^2+65*q^14)
       -867*q^10*w^10 
polynomial: 
           65*s^17+768*s^15+2808*s^13+2079*s^11-867*s^10-1286*s^9-585*s^7
                  +181*s^5+65*s^3 
factors= 
        s^3*(65*s^14+768*s^12+2808*s^10+2079*s^8-867*s^7-1286*s^6-585*s^4
                    +181*s^2+65) 
number of roots >0: 0 
poly(2)= 59331624 
=== 
k= 4 
coeff(term3, x, 4 )= 
               351*v^6-489*u*v^5-2058*u^2*v^4+546*u^3*v^3+4437*u^4*v^2
                      +2088*u^5*v+351*u^6 
coeff(lb, t, 4 )= -152*u^5*v^5 
wdiff= 
      q^4*w^4
         *(351*w^12-489*q^2*w^10-2058*q^4*w^8+546*q^6*w^6+4437*q^8*w^4
                   +2088*q^10*w^2+351*q^12)
       +152*q^10*w^10 
polynomial: 351*s^16+2088*s^14+4437*s^12+698*s^10-2058*s^8-489*s^6+351*s^4 
factors= s^4*(351*s^12+2088*s^10+4437*s^8+698*s^6-2058*s^4-489*s^2+351) 
number of roots >0: 0 
poly(2)= 75549104 
=== 
k= 5 
coeff(term3, x, 5 )= 
               741*v^5-2178*u*v^4-1476*u^2*v^3+3504*u^3*v^2+2997*u^4*v+741*u^5

coeff(lb, t, 5 )= -632*u^5*v^5 
wdiff= 
      q^5*w^5
         *(741*w^10-2178*q^2*w^8-1476*q^4*w^6+3504*q^6*w^4+2997*q^8*w^2
                   +741*q^10)
       +632*q^10*w^10 
polynomial: 741*s^15+2997*s^13+3504*s^11+632*s^10-1476*s^9-2178*s^7+741*s^5 
factors= s^5*(741*s^10+2997*s^8+3504*s^6+632*s^5-1476*s^4-2178*s^2+741) 
number of roots >0: 0 
poly(2)= 55645088 
=== 
k= 6 
coeff(term3, x, 6 )= 754*v^4-2698*u*v^3+1170*u^2*v^2+2412*u^3*v+754*u^4 
coeff(lb, t, 6 )= 2*u^5*v^5 
wdiff= 
      q^6*w^6*(754*w^8-2698*q^2*w^6+1170*q^4*w^4+2412*q^6*w^2+754*q^8)
       -2*q^10*w^10 
polynomial: 754*s^14+2412*s^12+1168*s^10-2698*s^8+754*s^6 
factors= 2*s^6*(377*s^8+1206*s^6+584*s^4-1349*s^2+377) 
number of roots >0: 0 
poly(2)= 22786688 
=== 
k= 7 
coeff(term3, x, 7 )= 390*v^3-1056*u*v^2+1134*u^2*v+390*u^3 
coeff(lb, t, 7 )= 531*u^5*v^5 
wdiff= q^7*w^7*(390*w^6-1056*q^2*w^4+1134*q^4*w^2+390*q^6)-531*q^10*w^10 
polynomial: 390*s^13+1134*s^11-531*s^10-1056*s^9+390*s^7 
factors= 3*s^7*(130*s^6+378*s^4-177*s^3-352*s^2+130) 
number of roots >0: 0 
poly(2)= 4482816 
=== 
k= 8 
coeff(term3, x, 8 )= 156*v^2-156*u*v+156*u^2 
coeff(lb, t, 8 )= 156*u^5*v^5 
wdiff= q^8*w^8*(156*w^4-156*q^2*w^2+156*q^4)-156*q^10*w^10 
polynomial: 156*s^12-312*s^10+156*s^8 
factors= 156*(s-1)^2*s^8*(s+1)^2 
number of roots >0: 2 
poly(2)= 359424 
(%o23) done
(%i24) "finally we proof that the lowerbbound polynomial has no positive root and that 

it is greater than 0 for t=1. Therefor it is greater or equal than 0 for all admissible values"
(%o24) "finally we proof that the lowerbbound polynomial has no positive root and that 

it is greater than 0 for t=1. Therefor it is greater or equal than 0 for all admissible values"
(%i25) poly:ratcoef(lowerbound,u^5*v^5)
(%o25) 156*t^8+531*t^7+2*t^6-632*t^5-152*t^4+867*t^3+834*t^2+299*t+40
(%i26) ev(poly,t = 1)
(%o26) 1945
(%i27) nroots(poly,0,inf)
(%o27) 0
(%i28) 

Here we list the coefficient functions so we can compare them to @MichaelRozenbergs function to see they are the same.

$$\begin{array}{r} \tag{1} \left(25\,u^3\,v^7-40\,u^4\,v^6+40\,u^6\,v^4+40\,u^7\,v^3\right)\,x^0 \\ \left(75\,u^2\,v^7-25\,u^3\,v^6-240\,u^4\,v^5+240\,u^5\,v^4+376\,u^ 6\,v^3+120\,u^7\,v^2\right)\,x^1 \\ \left(75\,u\,v^7+165\,u^2\,v^6-675\,u^3\,v^5+1416\,u^5\,v^3+888\,u^ 6\,v^2+120\,u^7\,v\right)\,x^2 \\ \left(65\,v^7+181\,u\,v^6-585\,u^2\,v^5-1286\,u^3\,v^4+2079\,u^4\,v ^3+2808\,u^5\,v^2+768\,u^6\,v+65\,u^7\right)\,x^3 \\ \left(351\,v^6-489\,u\,v^5-2058\,u^2\,v^4+546\,u^3\,v^3+4437\,u^4\, v^2+2088\,u^5\,v+351\,u^6\right)\,x^4 \\ \left(741\,v^5-2178\,u\,v^4-1476\,u^2\,v^3+3504\,u^3\,v^2+2997\,u^4 \,v+741\,u^5\right)\,x^5 \\ \left(754\,v^4-2698\,u\,v^3+1170\,u^2\,v^2+2412\,u^3\,v+754\,u^4 \right)\,x^6 \\ \left(390\,v^3-1056\,u\,v^2+1134\,u^2\,v+390\,u^3\right)\,x^7 \\ \left(156\,v^2-156\,u\,v+156\,u^2\right)\,x^8 \end{array}$$

To proof that this function is larger than $$\left(156\,t^8+531\,t^7+2\,t^6-632\,t^5-152\,t^4+867\,t^3+834\,t^2+ 299\,t+40\right)\,u^5\,v^5 \tag{2}$$ Rozenbergs's lower bound when we substitute $x$ by $t\sqrt(uv)$ we show that each coefficient of the polynomial $(1)$ is larger than the corresponding coefficient of the lower bound polynomial $(2)$. Then we show that the polynomial $(2)$ is larger than $0$ for all nonnegative $u$, $v$ and $t$. Details can be found in the Maxima script.

Instead of the Maxima nroots function, which is based on Sturm sequences, one could solve the equations by some numeric functions to see if there are zeros greater than zeros, e.g. calculating the roots of poly for k=7 gives the following:

(%i29) allroots(390*s^13+1134*s^11-531*s^10-1056*s^9+390*s^7 ,s);
(%o29) [s = 0.0,s = 0.0,s = 0.0,s = 0.0,s = 0.0,s = 0.0,s = 0.0,
        s = 0.007444635413686057*%i+0.7516683014652126,
        s = 0.7516683014652126-0.007444635413686057*%i,
        s = 0.3202741285237583*%i-0.6047586795035632,
        s = (-0.3202741285237583*%i)-0.6047586795035632,
        s = 1.93839678615644*%i-0.1469096219616494,
        s = (-1.93839678615644*%i)-0.1469096219616494]

So we can also conclude are no real roots greater than 0. But this method is not really acceptable if one does not analyze the impact of the rounding errors. And this can be very complicated. The nroots function works with integers (for integer polynomials) and so there are no rounding errors.

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I write a start for a full answer (this is an idea that @Starfall first proposed in comment). If someone wants to use it to end the proof, she/he is welcome!

Let $$f(x,y,z):=\frac{x^4}{ax^3+by^3}+\frac{y^4}{ay^3+bz^3}+\frac{z^4}{az^3+bx^3}.$$ Since $f$ is homogeneous of degree 1, it is sufficient to consider $x,y,z$ on the plane $P:=\{x+y+z=1\}$. Let $$g(x,y,z):=x+y+z-1$$ be the constraint function. We compute : $$\mathrm{d}f(x,y,z)=\left(\frac{ax^6+4bx^3y^3}{(ax^3+by^3)^2}-\frac{3bx^2z^4}{(az^3+bx^3)^2}\right)\mathrm{d}x+\left(\frac{ay^6+4by^3z^3}{(ay^3+bz^3)^2}-\frac{3bx^4y^2}{(ax^3+by^3)^2}\right)\mathrm{d}y$$ $$+\left(\frac{az^6+4bx^3z^3}{(az^3+bx^3)^2}-\frac{3by^4z^2}{(ay^3+bz^3)^2}\right)\mathrm{d}z,$$ $$\mathrm{d}g(x,y,z)=\mathrm{d}x+\mathrm{d}y+\mathrm{d}z.$$ Define the $2\times 3$ matrix $$M:=\begin{pmatrix} \frac{\partial f}{\partial x}(x,y,z) & \frac{\partial f}{\partial y}(x,y,z) & \frac{\partial f}{\partial z}(x,y,z)\\ \frac{\partial g}{\partial x}(x,y,z) & \frac{\partial g}{\partial y}(x,y,z) & \frac{\partial g}{\partial z}(x,y,z) \end{pmatrix}.$$ By Lagrange multipliers theorem, all the 3 sub-determinants of $M$ must vanish at a local minimum $(x,y,z)$ of $f$ on $P$.

Setting $$A:=ax^3+by^3,\quad B:=az^3+bx^3,\quad ay^3+bz^3,$$ cancelling the 3 sub-determinants of $M$ yields : \begin{align} \begin{cases} B^2C^2(ax^6+4bx^3y^3+3bx^4y^2)-3A^2C^2bx^2z^4-A^2B^2(ay^6+4by^3z^3)&=0\\ B^2C^2(ax^6+4bx^3y^3)-A^2C^2(3bx^2z^4+az^6+4bx^3z^3)+3A^2B^2by^4z^2&=0\\ A^2B^2(ay^6+4by^3z^3+3by^4z^2)-3B^2C^2bx^4y^2-A^2C^2(az^6+4bx^3z^3)&=0\\ x+y+z=1,\ x,y,z>0 \end{cases}. \end{align} Labelling the lines $(1)$, $(2)$, $(3)$ and $(4)$, we can see that $(1)-(2)=-(3)$, so that we can forget one of the three first lines.

Here we need to do some (boring) algebra, using the constraints of the fourth line above and maybe some tricks like writing $ax^3=A-by^3$ and $bx^4=(1-y-z)(B-az^3)$. But I am too busy now to try this, and I don't know if I would try later...

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  • $\begingroup$ Thank you very much for using Lagrange Multiplier ! One up vote from me :) $\endgroup$ – HN_NH May 24 '16 at 6:06
  • 1
    $\begingroup$ @HN_NH Thank you! Unfortunately, using the Lagrange multipliers theorem just yields a horrible system of algebraic equations. You should be able to solve it with a quite powerful program. $\endgroup$ – Nicolas May 24 '16 at 9:07
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(I consider it pertinent to add this answer to this challenging problem because the answer given here is different and simpler than the others given).

Lemma.-For every pair of positive $x,y$ it is verified that $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5x^3}\ge\frac{x+y}{13}$$ Proof.- It is an easy algebraic operation leading to $$(x^3-y^3)(x^2-y^2)\ge0$$ $$*******$$ Now, let $x,y,z$ be three arbitrary positive numbers. We have $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5x^3}\ge\frac{x+y}{13}\\\frac{x^4}{8x^3+5z^3}+\frac{z^4}{8z^3+5x^3}\ge\frac{x+z}{13}\\\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5y^3}\ge\frac{y+z}{13}$$ It follows that the sum $$\left(\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3}\right)+\left(\frac{x^4}{8x^3+5z^3}+\frac{z^4}{8z^3+5y^3}+\frac{y^4}{8y^3+5x^3}\right)$$ is greater or equal than $$\frac{2(x+y+z)}{13}$$ These two expressions in parentheses can not be both smaller than $\dfrac{x+y+z}{13}$ because their sum is greater than or equal to the double of this quantity. Thus one of them, say the first, satisfies $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3}\ge\frac{x+y+z}{13}$$ Now the function $f:\mathbb R^3_+\rightarrow \mathbb R_+$ defined by $f(x,y,z)=LHS$ is clearly continuous (it is even differentiable).

Choosing $(x,y,z)$ such that $0\lt||(x,y,z)-(x,z,y)||\to 0$ in $\mathbb R^3_+$ we have that $|f(x,y,z)-f(x,z,y)|\to 0$ in $\mathbb R_+$. This ensures that if $f(x,y,z)\ge\dfrac{x+y+z}{13}$ then this is so for $f(x,z,y)$ when $(x,z,y)$ is near enough of $(x,y,z)$. This ends the proof because from this we can deduce that one of the two expressions in parentheses above cannot be less to $\dfrac{x+y+z}{13}$ if the other is greater or equal than $\dfrac{x+y+z}{13}$ (which has been proved above).

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  • 1
    $\begingroup$ By your method I proved that $x^3y^2+y^3z^2+y^3z^2\geq(x^2+y^2+z^2)xyz$ for positives $x$, $y$ and $z$. Indeed, after summing last inequality with $x^3z^2+y^3x^2+z^3y^2\geq(x^2+y^2+z^2)xyz$ we obtain $\sum\limits_{cyc}(x^3y^2+y^3z^2)\geq2\sum\limits_{cyc}x^3yz,$ which is obvious by Muirhead. But first inequality is wrong. Try $x=7$, $y=1$ and $z=2$. If your reasoning is true we got a contradiction in Math. $\endgroup$ – Michael Rozenberg Aug 26 '18 at 1:47
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    $\begingroup$ why is this upvoted. When you are taking a limit, you are constantly changing the variables. $\endgroup$ – dezdichado Feb 24 at 22:24
  • $\begingroup$ Let $a,b,c \gt 0$ arbitraries and asked if $(a,b,c)$ satisfy $(*)$ where $(*)$ is the inequality $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$ According to the proven lemma we have $$\frac{a^4}{8a^3+5b^3}+\frac{b^4}{8b^3+5a^3}\ge\frac{a+b}{13}\\\frac{a^4}{8a^3+5c^3}+\frac{c^4}{8c^3+5a^3}\ge\frac{a+c}{13}\\\frac{b^4}{8b^3+5c^3}+\frac{c^4}{8c^3+5b^3}\ge\frac{b+c}{13}$$ From which the triplet $(a,b,c)$ satisfies $(*)$ $\endgroup$ – Piquito Feb 27 at 13:54
  • $\begingroup$ @Piquito: Unless I am mistaken, that only shows that one of (a, b, c) or (c, b, a) satisfies (*). $\endgroup$ – Martin R Mar 1 at 6:28
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At the beginning we transform the original inequality: $$\dfrac{8x^4}{8x^3+5y^3}+\dfrac{8y^4}{8y^3+5z^3}+\dfrac{8z^4}{8z^3+5x^3}\geq \dfrac8{13}(x+y+z),$$ $$x-\dfrac{5xy^3}{8x^3+5y^3}+y-\dfrac{5yz^3}{8y^3+5z^3}+z-\dfrac{5zx^3}{8z^3+5x^3}\geq \dfrac8{13}(x+y+z),$$ $$\dfrac{xy^3}{8x^3+5y^3}+\dfrac{yz^3}{8y^3+5z^3}+\dfrac{zx^3}{8z^3+5x^3}\leq \dfrac1{13}(x+y+z).\qquad(1)$$ Inequality $(1)$ is homogenius, and its decisions are determined up to an arbitrary common factor. We shall consider as the basic solution with $$x+y+z=1.$$ It is easy to see that among the unknowns $x, y, z$ in $(1)$ only one can have zero limit. Let $z\to 0,$ then inequality transforms to $$\dfrac{xy^3}{8x^3+5y^3}\leq\dfrac1{13},\quad x+y=1,$$ $$5y^3+8(1-y)^3-13y^3(1-y)\geq 0.$$ Note that $$5y^3+8(1-y)^3-13y^3(1-y) = 13y^4-16y^3+24 y^2-24y+8 \\= 4y(1-y)^3 + 17y^4-28 y^3+36y^2-28y+8 = 4y(1-y)^3+7(1-y)^4 + 10y^4-6y^2+1 = 4y(1-y)^3+7(1-y)^4+(3y^2-1)^2+y^4 \geq 0,$$ so the inequality is proved for this case .

For the case of $$x\not=0,\quad y\not=0,\quad z\not=0$$ we can use new method - of limiting transition.

I am not sure in its correctness, but discussion can be useful.

Let $$f(t)=\dfrac{t^3}{8+5t^3},$$ $$a = xf\left(\dfrac{y}x\right) + yf\left(\dfrac{z}y\right) + zf\left(\dfrac{x}z\right),$$ and find maximal possible value for $a.$

Using Cauchy-Schwarz inequality, we have $$\left(\sqrt{x}\cdot\sqrt{x}f\left(\dfrac{y}x\right)+\sqrt{y}\cdot\sqrt{y}f\left(\dfrac{z}y\right)+\sqrt{z}\cdot\sqrt{z}f\left(\dfrac{x}z\right)\right)^2 \leq (x+y+z)\left(xf^2\left(\dfrac{y}x\right)+yf^2\left(\dfrac{z}y\right)+zf^2\left(\dfrac{x}z\right)\right),$$ or $$a^2 \leq xf^2\left(\dfrac{y}x\right)+yf^2\left(\dfrac{z}y\right)+zf^2\left(\dfrac{x}z\right),$$ and similarly $$a^4 \leq xf^4\left(\dfrac{y}x\right) + yf^4\left(\dfrac{z}y\right) + zf^4\left(\dfrac{x}z\right),\dots$$ $$a^{2^n} \leq xf^{2^n}\left(\dfrac{y}x\right) + yf^{2^n}\left(\dfrac{z}y\right)+zf^{2^n}\left(\dfrac{x}z\right)\qquad(2)$$ for any $n\in\mathbb N.$

Let $f_m = \max\left\{f\left(\dfrac{y}x\right), f\left(\dfrac{z}y\right), f\left(\dfrac{x}z\right)\right\},$ then $(2)$ is equivalent to $$\left(\dfrac{a}{f_m}\right)^{2^n} \leq x\left(\dfrac{f\left(\dfrac{y}x\right)}{f_m}\right)^{2^n} + y\left(\dfrac{f\left(\dfrac{z}y\right)}{f_m}\right)^{2^n} + z\left(\dfrac{f\left(\dfrac{x}z\right)}{f_m}\right)^{2^n}.\qquad(3)$$

Inequality $(3)$ can be equality only if $$\lim_{n\to\infty}\left(\dfrac{a}{f_m}\right)^{2^n} = \lim_{n\to\infty}RHS.$$ If $$f\left(\dfrac{y}x\right)\not=f\left(\dfrac{z}y\right),\quad f\left(\dfrac{z}y\right)\not=f\left(\dfrac{x}z\right),\quad f\left(\dfrac{x}z\right)\not=f\left(\dfrac{y}x\right)$$ and for definiteness $$f_m=f\left(\dfrac{y}x\right),$$ then for $a,x,y,z>0$ the system of conditions becomes inconsistent: $$a=f_m,\quad x=1,\quad z=0.$$ If $$f_m=f\left(\dfrac{y}x\right)=f\left(\dfrac{z}y\right)\not=f\left(\dfrac{x}z\right),$$ or $$f_m=f\left(\dfrac{z}y\right)=f\left(\dfrac{x}z\right)\not=f\left(\dfrac{y}x\right),$$ or $$f_m=f\left(\dfrac{x}z\right)=f\left(\dfrac{y}x\right)\not=f\left(\dfrac{z}y\right),$$ one of the unknowns becomes zero, which reduces the problem to the case of the edges of area.

So remains the case of $$f_m = f\left(\dfrac{y}x\right) = f\left(\dfrac{z}y\right) = f\left(\dfrac{x}z\right),$$ $$\dfrac{y^3}{8x^3+5y^3}=\dfrac{z^3}{8y^3+5z^3}=\dfrac{x^3}{8z^3+5x^3},$$ or $$5+8\frac{x^3}{y^3} = 5+8\frac{y^3}{z^3} = 5+8\frac{z^3}{x^3},$$ $$\dfrac xy = \dfrac yz = \dfrac zx, \quad \dfrac{y}x\cdot\dfrac{z}y\cdot\dfrac{x}z = 1, $$ $$x=y=z,$$ $$a\leq f(1) = \dfrac1{13},$$

$$\boxed{\dfrac{x^4}{8x^3+5y^3}+\dfrac{y^4}{8y^3+5z^3}+\dfrac{z^4}{8z^3+5x^3}\geq \dfrac{x+y+z}{13}}.$$

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  • $\begingroup$ Dear @Yuri Negometyanov I don't see a connection between your paper and starting inequality. I am sorry. $\endgroup$ – Michael Rozenberg Jun 22 '17 at 2:40
  • $\begingroup$ @MichaelRozenberg More detail, please $\endgroup$ – Yuri Negometyanov Jun 22 '17 at 6:24
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Not sure, if I missed out anything here. Take a look.

For non negative, $X,Y,Z$, We can perhaps use Titu's inequality (a mix of Holder and CS), sometimes called Titu's screw lemma (https://en.wikipedia.org/wiki/Nesbitt%27s_inequality). \begin{equation} \sum_{k=1}^{n}{\frac{x_{k}^{2}}{a_{k}}} \ge \frac{\left(\sum_{k=1}^{n}{x_{k}}\right)^{2}}{\sum_{k=1}^{n}{a_{k}}} \end{equation}

With $n\to3$ terms, $x_{1}\to X^{2},x_{2} \to Y^{2}, x_{3} \to Z^{2}$ and $a_{1} \to A, a_{2}\to B, a_{3} \to C$, we will have

\begin{eqnarray*} \frac{\left(X^2\right)^{2}}{A}+\frac{\left(Y^2\right)^{2}}{B}+\frac{\left(Z^2\right)^{2}}{C} &\ge& \frac{\left(X^{2}+Y^{2}+Z^{2}\right)^{2}}{A+B+C} \\ \end{eqnarray*}

With \begin{eqnarray*} A &=& \alpha X^{3} +\beta Y^{3} \\ B &=& \alpha Y^{3} +\beta Z^{3} \\ C &=& \alpha Z^{3} +\beta X^{3} \end{eqnarray*}

where, \begin{eqnarray*} A+B+C &=& (\alpha+\beta) \left(X^{3} + Y^3+Z^3 \right) \end{eqnarray*}

\begin{eqnarray} \frac{X^4}{A}+\frac{Y^4}{B}+\frac{Z^4}{C} &=&\frac{\left(X^2\right)^{2}}{A}+\frac{\left(Y^2\right)^{2}}{B}+\frac{\left(Z^2\right)^{2}}{C}\\ &\ge& \frac{\left(X^{2}+Y^{2}+Z^{2}\right)^{2}}{A+B+C} \\ &=& \frac{\left(X^{2}+Y^{2}+Z^{2}\right)^{2}}{(\alpha+\beta) \left(X^{3} + Y^3+Z^3 \right)} \\ &\overset{(p)}{\ge}& \frac{\left(X^{3}+Y^{3}+Z^{3}\right)\left(X+Y+Z\right)}{(\alpha+\beta) \left(X^{3} + Y^3+Z^3 \right)} \\ &=& \frac{\left(X+Y+Z\right)}{(\alpha+\beta)} \end{eqnarray}

QED.

Here $(p)$ is from the fact that,

\begin{eqnarray*} (X^2+Y^2+Z^2)^{2} -\left(X^{3}+Y^{3}+Z^{3} \right) (X+Y+Z) &=& XY(X-Y)^{2}+YZ(Y-Z)^{2}+ZX(Z-X)^{2} \\ &\ge& 0 \end{eqnarray*}

Here $\alpha=8$ and $\beta=5$.

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  • $\begingroup$ By the way, I just used the name Titu (as it is known in the US IMO circle). I also hear names such as T2 Lemma, Engel's form, or Sedrakyan's inequality. It is indeed a special case of Cauchy-Schwarz. $\endgroup$ – NivPai Aug 22 '18 at 16:22
  • $\begingroup$ Yes, it is also known by the name Bergstrom’s inequality. There is also a slightly tighter generalization as well. $\endgroup$ – Rethna Pulikkoonattu Aug 22 '18 at 16:35
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    $\begingroup$ I think the identity for (p) is wrong: $(1^2+1^2+2^2)^2-(1^3+1^3+2^3)(1+1+2)=36-40=-4$. When you expand, you should get $2X^2Y^2-X^3Y-XY^3=-XY(X-Y)^2.$ $\endgroup$ – Jose Brox Aug 22 '18 at 16:37
  • $\begingroup$ Thanks @Jose Brox. Yes, then there is a hole in my argument. We still have it open then:-) $\endgroup$ – NivPai Aug 22 '18 at 17:04
  • $\begingroup$ @ Michael Rozenberg, I didn't mean so:-) I only meant to say, using this method:-). I still have feel that, there is a way to prove using T2 (Mostly, we may have to include the additional term involving max and then show that it stays positive). Will take a look over the weekend. $\endgroup$ – NivPai Aug 22 '18 at 17:20
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Let's reform this inequality in a way such that we can comprehend it better. Define $a=\dfrac{y}{x}$ and $b=\dfrac{z}{y}$, therefore $\dfrac{x}{z}={1\over ab}$. We can suume without lose of generality that $a,b\le1$ We need to prove that $$\dfrac{x}{8+5\left(\dfrac{y}{x}\right)^3}+\dfrac{y}{8+5\left(\dfrac{z}{y}\right)^3}+\dfrac{z}{8+5\left(\dfrac{x}{z}\right)^3}\ge\dfrac{x+y+z}{13}$$by dividing the two sides of the inequality by $x$ and substituting $a,b,c$ we have that$$\dfrac{1}{8+5\left(\dfrac{y}{x}\right)^3}+\dfrac{\dfrac{y}{x}}{8+5\left(\dfrac{z}{y}\right)^3}+\dfrac{\dfrac{z}{y}}{8+5\left(\dfrac{x}{z}\right)^3}\ge\dfrac{1+\dfrac{y}{x}+\dfrac{z}{x}}{13}$$and $$\dfrac{1}{8+5a^3}+\dfrac{a}{8+5b^3}+\dfrac{a^4b^4}{5+8a^3b^3}\ge \dfrac{1}{13}+\dfrac{a}{13}+\dfrac{ab}{13}$$which is equivalent to $$\left(\dfrac{1}{8+5a^3}-\dfrac{1}{13}\right)+\left(\dfrac{a}{8+5b^3}-\dfrac{a}{13}\right)+\left(\dfrac{a^4b^4}{5+8a^3b^3}-\dfrac{ab}{13}\right)\ge 0$$by simplifying each of the components and multiplying both sides in $\dfrac{13}{5}$ we obtain$$\dfrac{1-a^3}{8+5a^3}+\dfrac{a(1-b^3)}{8+5b^3}+\dfrac{a^4b^4-ab}{5+8a^3b^3}\ge0$$below is a depiction of $f(a,b)=\dfrac{1-a^3}{8+5a^3}+\dfrac{a(1-b^3)}{8+5b^3}+\dfrac{a^4b^4-ab}{5+8a^3b^3}$ for $0\le a,b\le 1$

enter image description here

which proves the inequality graphically (I believe that Lagrange multipliers or any other method based on 1st order derivations may help but i hadn't much time to think on it hope you find an analytic way) but neither such a time i spent on the problem nor a computer is given us in the exam :) also i appreciate if any one updates his/her comment with such an analytical method. I'm really curious about that.....

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For checking purposes.

Making $y = \lambda, \ z = \mu x$ and substituting into

$$ f(x,y,z) = \frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} - \frac{x+y+z}{13} $$

giving

$$ g(x,\lambda,\mu) =x\left( \frac{1}{5 \lambda ^3+8}+\frac{\lambda ^4}{8 \lambda ^3+5 \mu ^3}+\frac{\mu ^4}{8 \mu ^3+5}-\frac{1}{13} (\lambda +\mu +1)\right) $$

and discarding $x > 0$ we get

$$ \mathcal{G}(\lambda,\mu) = \frac{1}{5 \lambda ^3+8}+\frac{\lambda ^4}{8 \lambda ^3+5 \mu ^3}+\frac{\mu ^4}{8 \mu ^3+5}-\frac{1}{13} (\lambda +\mu +1) $$

Now solving the stationary conditions

$$ \nabla\mathcal{G}(\lambda,\mu) = 0 $$

we have the feasible stationary points with qualification.

$$ \left[ \begin{array}{cccl} \lambda & \mu & \mathcal{G}(\lambda,\mu) & \mbox{kind} \\ 1. & 1. & 0. & \mbox{min} \\ 0.485435 & 0.715221 & 0.000622453 & \mbox{min}\\ 0.646265 & 0.811309 & 0.000758688 & \mbox{saddle} \\ 1.37554 & 0.688678 & 0.000863479 & \mbox{min} \\ 1.25 & 0.77611 & 0.000941355 & \mbox{saddle} \\ 1.38778 & 1.85522 & 0.00123052 & \mbox{min} \\ 1.34211 & 1.74761 & 0.00123288 & \mbox{saddle} \\ \end{array} \right] $$

so the best solution is at $x = y = z = 1$

Attached the level contours for $\mathcal{G}(\lambda,\mu)$ with the stationary points in red.

enter image description here

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This is too long to fit into a comment. I wanted to ask a question about my proof on this problem. (It might help discover another proof)


This proof has a flaw -- From $AB \ge C$ and $A \ge D$, I wrongly implied that $DB \ge C$.

Is there a way to slightly modify it such that it can prove the statement or is it completely wrong?


Seeing that the inequality is homogeneous (meaning that the transformation $(x, y, z) \mapsto (kx, ky, kz)$ does not change anything), it is natural to impose a constraint on it. So let us assume without the loss of generality that $xyz=1$.

From Cauchy-Schwarz Inequality,

$$([8x^3+5y^3]+[8y^3+5z^3]+[8z^3+5x^3])(\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3})\geqslant (x^2+y^2+z^2)^2$$

Since (By AM-GM) $$[8x^3+5y^3]+[8y^3+5z^3]+[8z^3+5x^3] = 13(x^3+y^3+z^3) \geqslant 13(3 \sqrt[3]{(xyz)^3}) = 13(3)$$

Therefore

$([8x^3+5y^3]+[8y^3+5z^3]+[8z^3+5x^3])(\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3}) \geqslant (13)(3)(\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3}) \geqslant (x^2+y^2+z^2)^2$

Therefore

$$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{(x^2+y^2+z^2)^2}{(13)(3)}$$

Now it reamins to prove that $\frac{(x^2+y^2+z^2)^2}{(13)(3)} \geqslant \frac{x+y+z}{13}$, i.e.

$$(x^2+y^2+z^2)(x^2+y^2+z^2)\geqslant 3(x+y+z)$$

which is straightforward by AM-GM:

Notice that for all $xyz=1$

$$(x - 1)^2 + (y-1)^2 + (z - 1)^2 \ge 0$$ $$x^2 + y^2 + z^2 - 2a - 2b - 2c + 3 \ge 0$$ $$x^2 + y^2 + z^2 \ge -3 + (x + y + z) + (x + y + z)$$

But by AM-GM, $x + y + z \ge 3\sqrt[3]{xyz} = 3$. So, $$x^2 + y^2 + z^2 \ge -3 + 3 + (x + y + z)$$ $$x^2 + y^2 + z^2 \ge x + y + z \ge 3$$

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  • $\begingroup$ Your inequality $\sum\limits_{cyc}\frac{x^4}{8x^3+5y^3}\geq\frac{(x^2+y^2+z^2)^2}{(13)(3)}$ it's $\sum\limits_{cyc}\frac{x^4}{8x^3+5y^3}\geq\frac{(x^2+y^2+z^2)^2}{39xyz}$. I think the last inequality is wrong. Try $x=y=1$ and $z=\frac{1}{2}.$ $\endgroup$ – Michael Rozenberg Aug 26 '18 at 1:28
  • $\begingroup$ quoting from the first paragraph: Seeing that the inequality is homogeneous (meaning that the transformation $(x, y, z) \mapsto (kx, ky, kz)$ does not change anything), it is natural to impose a constraint on it. So let us assume without the loss of generality that $xyz=1$. $\endgroup$ – Vee Hua Zhi Aug 26 '18 at 2:04
  • $\begingroup$ If you understood it try to check my counterexample. I'll write again: The inequality $\sum\limits_{cyc}\frac{x^4}{8x^3+5y^3}\geq\frac{(x^2+y^2+z^2)^2}{(13)(3)}$ is wrong. Try $x=4$ and $y=z=0.5$. $\endgroup$ – Michael Rozenberg Aug 26 '18 at 6:35
  • $\begingroup$ OK I get it @MichaelRozenberg $\endgroup$ – Vee Hua Zhi Aug 28 '18 at 12:47
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I have finally found a solution . In fact we start to study the 2 variables version of this inequality we have :

$$\frac{a^4}{8a^3+5b^3}+\frac{b^4}{8b^3+5a^3}\geq \frac{a+b}{13}$$

Proof:

We have with $x=\frac{a}{b}$ : $$\frac{x^4}{8x^3+5}+\frac{1}{8+5x^3}\geq \frac{1+x}{13}$$ Or $$\frac{5}{13}(x - 1)^2 (x + 1) (x^2 + x + 1) (5 x^2 - 8 x + 5)\geq 0$$

So we have (if we permute the variables $a,b,c$ and add the three inequalities ) :

$$\sum_{cyc}\frac{a^4}{8a^3+5b^3}+\sum_{cyc}\frac{a^4}{8a^3+5c^3}\geq \frac{a+b+c}{6.5}$$

If we have $\sum_{cyc}\frac{a^4}{8a^3+5b^3}\geq\sum_{cyc}\frac{a^4}{8a^3+5c^3}$

We have : $$\sum_{cyc}\frac{a^4}{8a^3+5b^3}\geq \frac{a+b+c}{13}$$ But also $$\frac{(a-\epsilon)^4}{8(a-\epsilon)^3+5b^3}+\frac{(b)^4}{8(b)^3+5(c+\epsilon)^3}+\frac{(c+\epsilon)^4}{8(c+\epsilon)^3+5(a-\epsilon)^3}\geq \frac{a+b+c}{13}$$ If we put $a\geq c $ and $\epsilon=a-c$

We finally obtain : $$\sum_{cyc}\frac{a^4}{8a^3+5c^3}\geq \frac{a+b+c}{13}$$

If we have $\sum_{cyc}\frac{a^4}{8a^3+5b^3}\leq\sum_{cyc}\frac{a^4}{8a^3+5c^3}$

The proof is the same as above .

So all the cases are present so it's proved !

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