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Notebook[{


Cell[CellGroupData[{Cell[TextData[
"showproof = True;\nshowHeuristics\t= False;\n\n\ntest[x^2+y^2>=2*x*y];\n\
test[x^3*y+x*y^3<=x^4+y^4];\ntest[(x+y)*(y+z)*(z+x)>=8*x*y*z];\n\
test[4*(x^3+y^3)>=(x+y)^3];\ntest[(x+y+z)^3>=27*x*y*z];\n\
test[x*y*z>=(x+y-z)*(y+z-x)*(z+x-y)];\ntest[(x^3+y^3)^2<=(x^2+y^2)^3];\n\
test[8*(x^3+y^3+z^3)^2>=9*(x^2+y*z)*(y^2+z*x)*(z^2+x*y)];\n\
test[4*(x^5+y^5+z^5+w^5)>=(x^3+y^3+z^3+w^3)*(x^2+y^2+z^2+w^2)];\n\
test[(b+c+d)*(c+d+a)*(d+a+b)*(a+b+c)>=81*a*b*c*d];\n\
test[(x^2*y+y^2*z+z^2*x)*(x*y^2+y*z^2+z*x^2)>=9*x^2*y^2*z^2];\n\
test[6*x*y*z<=x*y*(x+y)+y*z*(y+z)+z*x*(z+x)];\n\
test[x^2*y^2+y^2*z^2+z^2*x^2>=x*y*z*(x+y+z)];\n\
test[(x+y+z)^3>=(x+y-z)*(y+z-x)*(z+x-y)];\ntest[27*(x^4+y^4+z^4)>=(x+y+z)^4];\
\ntest[(x+y+z+w)*(x^3+y^3+z^3+w^3)>=(x^2+y^2+z^2+w^2)^2];\n\
test[x*(x-y)*(x-z)+y*(y-z)*(y-x)+z*(z-x)*(z-y)>=0];\n\
test[x^2*(x-y)*(x-z)+y^2*(y-z)*(y-x)+z^2*(z-x)*(z-y)>=0];\n\
test[x^5*(x-y)*(x-z)+y^5*(y-z)*(y-x)+z^5*(z-x)*(z-y)>=0];\n\
test[9*(x^6+y^6+z^6)>=(x^3+y^3+z^3)*(x^2+y^2+z^2)*(x+y+z)];\n\
test[16*(x^3+y^3+z^3+w^3)>=(x+y+z+w)^3];\n\
test[(x+y-z)^2+(y+z-x)^2+(z+x-y)^2>=x*y+y*z+z*x];\n\
test[x^3+y^3+z^3+3*x*y*z>=x^2*(y+z)+z^2*(x+y)+y^2*(z+x)];\n\
test[6*x*y*z<=x^2*(y+z)+y^2*(z+x)+z^2*(x+y)];\n\
test[x^2*(y+z)+y^2*(z+x)+z^2*(x+y)<=2*(x^3+y^3+z^3)];\n\
test[a*b*c*d>=(b+c+d-2*a)*(c+d+a-2*b)*(d+a+b-2*c)*(a+b+c-2*d)];\n\
test[3*(x^2*y+y^2*z+z^2*x)*(x*y^2+y*z^2+z*x^2)>=x*y*z*(x+y+z)^3];\n\
test[(x^7+y^7)^3<=(x^3+y^3)^7];\ntest[1/x+1/y+1/z>=9/(x+y+z)];\n\
test[x/(y+z)+y/(z+x)+z/(x+y)>=3/2];\n\
test[x^(-5)*(x-y)*(x-z)+y^(-5)*(y-z)*(y-x)+z^(-5)*(z-x)*(z-y)>=0];\n\
test[1/x+1/y+1/z<=(x^8+y^8+z^8)/(x^3*y^3*z^3)];\n\
test[1/(x+y+z)+1/(y+z+w)+1/(z+w+x)+1/(w+x+y)>=(16/3)/(x+y+z+w)];\n\
test[(x^2+y^2)/(x+y)+(y^2+z^2)/(y+z)+(z^2+x^2)/(z+x)>=x+y+z];\n\
test[(a^3+b^3+c^3+d^3+e^3)*(1/a+1/b+1/c+1/d+1/e)>=5*(a^2+b^2+c^2+d^2+e^2)];\n\
test[3/(b+c+d)+3/(c+d+a)+3/(d+a+b)+3/(a+b+c)>=16/(a+b+c+d)];\n\
test[(x^3+y^3+z^3+v^3+w^3)(1/x+1/y+1/z+1/v+1/w)>=\n5(x^2+y^2+z^2+v^2+w^2)];\n\
"], "Input",
  InitializationCell->True,
  AspectRatioFixed->True],

Cell[TextData[
" \n----------\n 2    2\nx  + y  >= 2 x y\nExpressing in terms of symmetric \
polynomials gives:\n[2,0] >= [1,1]\nThis result follows from the majorization \
theorem.\n \n----------\n 3        3     4    4\nx  y + x y  <= x  + y\n\
Expressing in terms of symmetric polynomials gives:\n[4,0] >= [3,1]\nThis \
result follows from the majorization theorem.\n \n----------\n(x + y) (x + z) \
(y + z) >= 8 x y z\nExpanding and collecting terms of the same sign gives:\n \
2        2    2      2        2      2\nx  y + x y  + x  z + y  z + x z  + y \
z  >= 6 x y z\nExpressing in terms of symmetric polynomials gives:\n[2,1,0] \
>= [1,1,1]\nThis result follows from the majorization theorem.\n \n----------\
\n    3    3            3\n4 (x  + y ) >= (x + y)\nExpanding and collecting \
terms of the same sign gives:\n   3      3       2          2\n3 x  + 3 y  >= \
3 x  y + 3 x y\nDividing both sides by 3 (x + y) gives:\n 2          2\nx  - \
x y + y  >= x y\nExpanding and collecting terms of the same sign gives:\n 2   \
 2\nx  + y  >= 2 x y\nExpressing in terms of symmetric polynomials gives:\n\
[2,0] >= [1,1]\nThis result follows from the majorization theorem.\n \n\
----------\n           3\n(x + y + z)  >= 27 x y z\nExpanding and collecting \
terms of the same sign gives:\n 3      2          2    3      2        2      \
    2        2\nx  + 3 x  y + 3 x y  + y  + 3 x  z + 3 y  z + 3 x z  + 3 y z  \
+ \n \n    3\n   z  >= 21 x y z\nExpressing in terms of symmetric polynomials \
gives:\n6 [2,1,0] + [3,0,0] >= 7 [1,1,1]\nThis follows from the following \
majorizations:\n[3,0,0] >= [1,1,1]\n6 [2,1,0] >= 6 [1,1,1]\n \n----------\nx \
y z >= (x + y - z) (x - y + z) (-x + y + z)\nExpanding and collecting terms \
of the same sign gives:\n 3    3              3     2        2    2      2    \
    2      2\nx  + y  + 3 x y z + z  >= x  y + x y  + x  z + y  z + x z  + y \
z\nExpressing in terms of symmetric polynomials gives:\n[1,1,1] + [3,0,0] >= \
2 [2,1,0]\nThis follows from the following majorizations:\n[1,1,1] + [3,0,0] \
>= 2 [2,1,0]\n \n----------\n  3    3 2      2    2 3\n(x  + y )  <= (x  + y \
)\nExpanding and collecting terms of the same sign gives:\n   4  2      2  4  \
     3  3\n3 x  y  + 3 x  y  >= 2 x  y\n                        2  2\n\
Dividing both sides by x  y  gives:\n   2      2\n3 x  + 3 y  >= 2 x y\n\
Expressing in terms of symmetric polynomials gives:\n3 [2,0] >= [1,1]\nThis \
result follows from the majorization theorem.\n \n----------\n    3    3    3 \
2        2          2                2\n8 (x  + y  + z )  >= 9 (y  + x z) (x  \
+ y z) (x y + z )\nExpanding and collecting terms of the same sign gives:\n   \
6      3  3      6      3  3      3  3      6\n8 x  + 7 x  y  + 8 y  + 7 x  z \
 + 7 y  z  + 8 z  >= \n \n     4            4         2  2  2          4\n  9 \
x  y z + 9 x y  z + 18 x  y  z  + 9 x y z\nExpressing in terms of symmetric \
polynomials gives:\n7 [3,3,0] + 8 [6,0,0] >= 6 [2,2,2] + 9 [4,1,1]\nThis \
follows from the following majorizations:\n8 [6,0,0] >= 8 [4,1,1]\n6 [3,3,0] \
>= 6 [2,2,2]\nleft  side: {7 [3,3,0], 8 [6,0,0]}\nright side: {6 [2,2,2], 9 \
[4,1,1]}\n********* COULD NOT PROVE THIS INEQUALITY *********\n \n----------\n\
    5    5    5    5       2    2    2    2    3    3    3    3\n4 (w  + x  + \
y  + z ) >= (w  + x  + y  + z ) (w  + x  + y  + z )\nExpanding and collecting \
terms of the same sign gives:\n   5      5      5      5\n3 w  + 3 x  + 3 y  \
+ 3 z  >= \n \n   3  2    2  3    3  2    3  2    2  3    2  3    3  2    3  \
2\n  w  x  + w  x  + w  y  + x  y  + w  y  + x  y  + w  z  + x  z  + \n \n    \
3  2    2  3    2  3    2  3\n   y  z  + w  z  + x  z  + y  z\nExpressing in \
terms of symmetric polynomials gives:\n[5,0,0,0] >= [3,2,0,0]\nThis result \
follows from the majorization theorem.\n \n----------\n(a + b + c) (a + b + \
d) (a + c + d) (b + c + d) >= 81 a b c d\nExpanding and collecting terms of \
the same sign gives:\n 3        2  2      3    3        2            2      3\
\na  b + 2 a  b  + a b  + a  c + 4 a  b c + 4 a b  c + b  c + \n \n      2  2 \
         2      2  2      3      3    3        2\n   2 a  c  + 4 a b c  + 2 b \
 c  + a c  + b c  + a  d + 4 a  b d + \n \n        2      3        2          \
2            2          2\n   4 a b  d + b  d + 4 a  c d + 4 b  c d + 4 a c  \
d + 4 b c  d + \n \n    3        2  2          2      2  2          2         \
 2\n   c  d + 2 a  d  + 4 a b d  + 2 b  d  + 4 a c d  + 4 b c d  + \n \n      \
2  2      3      3      3\n   2 c  d  + a d  + b d  + c d  >= 72 a b c d\n\
Expressing in terms of symmetric polynomials gives:\n4 [2,1,1,0] + [2,2,0,0] \
+ [3,1,0,0] >= 6 [1,1,1,1]\nThis follows from the following majorizations:\n\
[3,1,0,0] >= [1,1,1,1]\n[2,2,0,0] >= [1,1,1,1]\n4 [2,1,1,0] >= 4 [1,1,1,1]\n \
\n----------\n  2      2        2      2    2        2        2  2  2\n(x  y \
+ y  z + x z ) (x y  + x  z + y z ) >= 9 x  y  z\nExpanding and collecting \
terms of the same sign gives:\n 3  3    4          4      3  3    3  3        \
4       2  2  2\nx  y  + x  y z + x y  z + x  z  + y  z  + x y z  >= 6 x  y  \
z\nExpressing in terms of symmetric polynomials gives:\n[3,3,0] + [4,1,1] >= \
2 [2,2,2]\nThis follows from the following majorizations:\n[3,3,0] >= [2,2,2]\
\n[4,1,1] >= [2,2,2]\n \n----------\n6 x y z <= x y (x + y) + x z (x + z) + y \
z (y + z)\nExpressing in terms of symmetric polynomials gives:\n[2,1,0] >= \
[1,1,1]\nThis result follows from the majorization theorem.\n \n----------\n \
2  2    2  2    2  2\nx  y  + x  z  + y  z  >= x y z (x + y + z)\nExpressing \
in terms of symmetric polynomials gives:\n[2,2,0] >= [2,1,1]\nThis result \
follows from the majorization theorem.\n \n----------\n           3\n(x + y + \
z)  >= (x + y - z) (x - y + z) (-x + y + z)\nExpanding and collecting terms \
of the same sign gives:\n   3      2          2      3      2                 \
 2\n2 x  + 2 x  y + 2 x y  + 2 y  + 2 x  z + 8 x y z + 2 y  z + \n \n        \
2        2      3\n   2 x z  + 2 y z  + 2 z  >= 0\nThis is true because a sum \
of positive terms must be positive.\n \n----------\n     4    4    4          \
      4\n27 (x  + y  + z ) >= (x + y + z)\nExpanding and collecting terms of \
the same sign gives:\n    4       4       4       3        2  2        3      \
3\n26 x  + 26 y  + 26 z  >= 4 x  y + 6 x  y  + 4 x y  + 4 x  z + \n \n       \
2             2        3        2  2           2      2  2\n   12 x  y z + 12 \
x y  z + 4 y  z + 6 x  z  + 12 x y z  + 6 y  z  + \n \n        3        3\n   \
4 x z  + 4 y z\nDividing both sides by 2 gives:\n    4       4       4       \
3        2  2        3      3\n13 x  + 13 y  + 13 z  >= 2 x  y + 3 x  y  + 2 \
x y  + 2 x  z + \n \n      2            2        3        2  2          2     \
 2  2\n   6 x  y z + 6 x y  z + 2 y  z + 3 x  z  + 6 x y z  + 3 y  z  + \n \n \
       3        3\n   2 x z  + 2 y z\nExpressing in terms of symmetric \
polynomials gives:\n13 [4,0,0] >= 6 [2,1,1] + 3 [2,2,0] + 4 [3,1,0]\nThis \
follows from the following majorizations:\n6 [4,0,0] >= 6 [2,1,1]\n3 [4,0,0] \
>= 3 [2,2,0]\n4 [4,0,0] >= 4 [3,1,0]\n \n----------\n                  3    3 \
   3    3       2    2    2    2 2\n(w + x + y + z) (w  + x  + y  + z ) >= (w \
 + x  + y  + z )\nExpanding and collecting terms of the same sign gives:\n 3  \
      3    3      3        3      3    3      3      3\nw  x + w x  + w  y + \
x  y + w y  + x y  + w  z + x  z + y  z + \n \n      3      3      3       2  \
2      2  2      2  2      2  2\n   w z  + x z  + y z  >= 2 w  x  + 2 w  y  + \
2 x  y  + 2 w  z  + \n \n      2  2      2  2\n   2 x  z  + 2 y  z\n\
Expressing in terms of symmetric polynomials gives:\n[3,1,0,0] >= [2,2,0,0]\n\
This result follows from the majorization theorem.\n \n----------\nx (x - y) \
(x - z) + y (-x + y) (y - z) + z (-x + z) (-y + z) >= 0\nExpanding and \
collecting terms of the same sign gives:\n 3    3              3     2        \
2    2      2        2      2\nx  + y  + 3 x y z + z  >= x  y + x y  + x  z + \
y  z + x z  + y z\nExpressing in terms of symmetric polynomials gives:\n\
[1,1,1] + [3,0,0] >= 2 [2,1,0]\nThis follows from the following \
majorizations:\n[1,1,1] + [3,0,0] >= 2 [2,1,0]\n \n----------\n 2             \
       2                     2\nx  (x - y) (x - z) + y  (-x + y) (y - z) + z  \
(-x + z) (-y + z) >= 0\nExpanding and collecting terms of the same sign \
gives:\n 4    4    2          2          2    4\nx  + y  + x  y z + x y  z + \
x y z  + z  >= \n \n   3        3    3      3        3      3\n  x  y + x y  \
+ x  z + y  z + x z  + y z\nExpressing in terms of symmetric polynomials \
gives:\n[2,1,1] + [4,0,0] >= 2 [3,1,0]\nThis follows from the following \
majorizations:\n[2,1,1] + [4,0,0] >= 2 [3,1,0]\n \n----------\n 5             \
       5                     5\nx  (x - y) (x - z) + y  (-x + y) (y - z) + z  \
(-x + z) (-y + z) >= 0\nExpanding and collecting terms of the same sign \
gives:\n 7    7    5          5          5    7\nx  + y  + x  y z + x y  z + \
x y z  + z  >= \n \n   6        6    6      6        6      6\n  x  y + x y  \
+ x  z + y  z + x z  + y z\nExpressing in terms of symmetric polynomials \
gives:\n[5,1,1] + [7,0,0] >= 2 [6,1,0]\nThis follows from the following \
majorizations:\n[5,1,1] + [7,0,0] >= 2 [6,1,0]\n \n----------\n    6    6    \
6                   2    2    2    3    3    3\n9 (x  + y  + z ) >= (x + y + \
z) (x  + y  + z ) (x  + y  + z )\nExpanding and collecting terms of the same \
sign gives:\n   6      6      6     5      4  2      3  3    2  4      5\n8 x \
 + 8 y  + 8 z  >= x  y + x  y  + 2 x  y  + x  y  + x y  + \n \n    5      3  \
2      2  3      5      4  2    3    2      3  2\n   x  z + x  y  z + x  y  z \
+ y  z + x  z  + x  y z  + x y  z  + \n \n    4  2      3  3    2    3      2 \
 3      3  3    2  4    2  4\n   y  z  + 2 x  z  + x  y z  + x y  z  + 2 y  z \
 + x  z  + y  z  + \n \n      5      5\n   x z  + y z\nExpressing in terms of \
symmetric polynomials gives:\n4 [6,0,0] >= [3,2,1] + [3,3,0] + [4,2,0] + \
[5,1,0]\nThis follows from the following majorizations:\n[6,0,0] >= [3,2,1]\n\
[6,0,0] >= [3,3,0]\n[6,0,0] >= [4,2,0]\n[6,0,0] >= [5,1,0]\n \n----------\n   \
  3    3    3    3                    3\n16 (w  + x  + y  + z ) >= (w + x + y \
+ z)\nExpanding and collecting terms of the same sign gives:\n    3       3   \
    3       3\n15 w  + 15 x  + 15 y  + 15 z  >= \n \n     2          2      2 \
                 2          2        2\n  3 w  x + 3 w x  + 3 w  y + 6 w x y \
+ 3 x  y + 3 w y  + 3 x y  + \n \n      2                  2                  \
          2\n   3 w  z + 6 w x z + 3 x  z + 6 w y z + 6 x y z + 3 y  z + \n \n\
        2        2        2\n   3 w z  + 3 x z  + 3 y z\nDividing both sides \
by 3 gives:\n   3      3      3      3\n5 w  + 5 x  + 5 y  + 5 z  >= \n \n   \
2        2    2                2        2      2    2\n  w  x + w x  + w  y + \
2 w x y + x  y + w y  + x y  + w  z + \n \n              2                    \
      2        2      2      2\n   2 w x z + x  z + 2 w y z + 2 x y z + y  z \
+ w z  + x z  + y z\nExpressing in terms of symmetric polynomials gives:\n5 \
[3,0,0,0] >= 2 [1,1,1,0] + 3 [2,1,0,0]\nThis follows from the following \
majorizations:\n2 [3,0,0,0] >= 2 [1,1,1,0]\n3 [3,0,0,0] >= 3 [2,1,0,0]\n \n\
----------\n           2              2               2\n(x + y - z)  + (x - \
y + z)  + (-x + y + z)  >= x y + x z + y z\nExpanding and collecting terms of \
the same sign gives:\n   2      2      2\n3 x  + 3 y  + 3 z  >= 3 x y + 3 x z \
+ 3 y z\nDividing both sides by 3 gives:\n 2    2    2\nx  + y  + z  >= x y + \
x z + y z\nExpressing in terms of symmetric polynomials gives:\n[2,0,0] >= \
[1,1,0]\nThis result follows from the majorization theorem.\n \n----------\n \
3    3              3             2    2            2\nx  + y  + 3 x y z + z  \
>= (x + y) z  + y  (x + z) + x  (y + z)\nExpressing in terms of symmetric \
polynomials gives:\n[1,1,1] + [3,0,0] >= 2 [2,1,0]\nThis follows from the \
following majorizations:\n[1,1,1] + [3,0,0] >= 2 [2,1,0]\n \n----------\n     \
               2    2            2\n6 x y z <= (x + y) z  + y  (x + z) + x  \
(y + z)\nExpressing in terms of symmetric polynomials gives:\n[2,1,0] >= \
[1,1,1]\nThis result follows from the majorization theorem.\n \n----------\n  \
       2    2            2                3    3    3\n(x + y) z  + y  (x + \
z) + x  (y + z) <= 2 (x  + y  + z )\nExpressing in terms of symmetric \
polynomials gives:\n[3,0,0] >= [2,1,0]\nThis result follows from the \
majorization theorem.\n \n----------\na b c d >= (a + b + c - 2 d) (a + b - 2 \
c + d) (a - 2 b + c + d) \n \n   (-2 a + b + c + d)\nExpanding and collecting \
terms of the same sign gives:\n   4      4      2            2            2   \
   4      2\n2 a  + 2 b  + 6 a  b c + 6 a b  c + 6 a b c  + 2 c  + 6 a  b d + \
\n \n        2        2          2            2          2\n   6 a b  d + 6 a \
 c d + 6 b  c d + 6 a c  d + 6 b c  d + \n \n          2          2          \
2      4\n   6 a b d  + 6 a c d  + 6 b c d  + 2 d  >= \n \n   3        2  2   \
   3    3      3        2  2      2  2      3\n  a  b + 6 a  b  + a b  + a  c \
+ b  c + 6 a  c  + 6 b  c  + a c  + \n \n      3    3      3                  \
 3        2  2      2  2\n   b c  + a  d + b  d + 32 a b c d + c  d + 6 a  d  \
+ 6 b  d  + \n \n      2  2      3      3      3\n   6 c  d  + a d  + b d  + \
c d\nExpressing in terms of symmetric polynomials gives:\n18 [2,1,1,0] + 2 \
[4,0,0,0] >= \n \n  8 [1,1,1,1] + 9 [2,2,0,0] + 3 [3,1,0,0]\nThis follows \
from the following majorizations:\n2 [4,0,0,0] >= 2 [2,2,0,0]\n8 [2,1,1,0] >= \
8 [1,1,1,1]\nleft  side: {18 [2,1,1,0], 2 [4,0,0,0]}\nright side: {8 \
[1,1,1,1], 9 [2,2,0,0], 3 [3,1,0,0]}\n********* COULD NOT PROVE THIS \
INEQUALITY *********\n \n----------\n    2      2        2      2    2        \
2                      3\n3 (x  y + y  z + x z ) (x y  + x  z + y z ) >= x y \
z (x + y + z)\nExpanding and collecting terms of the same sign gives:\n   3  \
3      4            4        2  2  2      3  3      3  3\n3 x  y  + 2 x  y z \
+ 2 x y  z + 3 x  y  z  + 3 x  z  + 3 y  z  + \n \n          4       3  2     \
   2  3        3    2        3  2\n   2 x y z  >= 3 x  y  z + 3 x  y  z + 3 x \
 y z  + 3 x y  z  + \n \n      2    3        2  3\n   3 x  y z  + 3 x y  z\n\
Expressing in terms of symmetric polynomials gives:\n[2,2,2] + 3 [3,3,0] + 2 \
[4,1,1] >= 6 [3,2,1]\nThis follows from the following majorizations:\n[2,2,2] \
+ [3,3,0] >= 2 [3,2,1]\n2 [4,1,1] >= 2 [3,2,1]\n2 [3,3,0] >= 2 [3,2,1]\n \n\
----------\n  7    7 3      3    3 7\n(x  + y )  <= (x  + y )\nExpanding and \
collecting terms of the same sign gives:\n   18  3       15  6       12  9    \
   9  12       6  15\n7 x   y  + 21 x   y  + 35 x   y  + 35 x  y   + 21 x  y  \
 + \n \n      3  18       14  7      7  14\n   7 x  y   >= 3 x   y  + 3 x  y\n\
                        3  3\nDividing both sides by x  y  (x + y) gives:\n   \
14      13        12  2       11  3       10  4       9  5\n7 x   - 7 x   y + \
7 x   y  + 14 x   y  - 14 x   y  + 14 x  y  + \n \n       8  6       7  7     \
  6  8       5  9       4  10\n   21 x  y  - 21 x  y  + 21 x  y  + 14 x  y  - \
14 x  y   + \n \n       3  11      2  12        13      14\n   14 x  y   + 7 \
x  y   - 7 x y   + 7 y   >= \n \n     10  4      9  5      8  6      7  7     \
 6  8      5  9\n  3 x   y  - 3 x  y  + 3 x  y  - 3 x  y  + 3 x  y  - 3 x  y  \
+ \n \n      4  10\n   3 x  y\nExpanding and collecting terms of the same \
sign gives:\n   14      12  2       11  3       9  5       8  6       6  8\n7 \
x   + 7 x   y  + 14 x   y  + 17 x  y  + 18 x  y  + 18 x  y  + \n \n       5  \
9       3  11      2  12      14\n   17 x  y  + 14 x  y   + 7 x  y   + 7 y   \
>= \n \n     13         10  4       7  7       4  10        13\n  7 x   y + \
17 x   y  + 18 x  y  + 17 x  y   + 7 x y\nExpressing in terms of symmetric \
polynomials gives:\n18 [8,6] + 17 [9,5] + 14 [11,3] + 7 [12,2] + 7 [14,0] >= \
\n \n  9 [7,7] + 17 [10,4] + 7 [13,1]\nThis follows from the following \
majorizations:\n7 [14,0] >= 7 [13,1]\n9 [8,6] >= 9 [7,7]\n7 [12,2] >= 7 \
[10,4]\n10 [11,3] >= 10 [10,4]\n \n----------\n1   1   1        9\n- + - + - \
>= ---------\nx   y   z    x + y + z\nMultiplying both sides by x y z (x + y \
+ z) gives:\n 2        2    2                2        2      2\nx  y + x y  + \
x  z + 3 x y z + y  z + x z  + y z  >= 9 x y z\nExpanding and collecting \
terms of the same sign gives:\n 2        2    2      2        2      2\nx  y \
+ x y  + x  z + y  z + x z  + y z  >= 6 x y z\nExpressing in terms of \
symmetric polynomials gives:\n[2,1,0] >= [1,1,1]\nThis result follows from \
the majorization theorem.\n \n----------\n  z       y       x      3\n----- + \
----- + ----- >= -\nx + y   x + z   y + z    2\nMultiplying both sides by 2 \
(x + y) (x + z) (y + z) gives:\n   3      2          2      3      2          \
        2\n2 x  + 2 x  y + 2 x y  + 2 y  + 2 x  z + 6 x y z + 2 y  z + \n \n  \
      2        2      3\n   2 x z  + 2 y z  + 2 z  >= \n \n     2          2  \
    2                  2          2        2\n  3 x  y + 3 x y  + 3 x  z + 6 \
x y z + 3 y  z + 3 x z  + 3 y z\nExpanding and collecting terms of the same \
sign gives:\n   3      3      3     2        2    2      2        2      2\n2 \
x  + 2 y  + 2 z  >= x  y + x y  + x  z + y  z + x z  + y z\nExpressing in \
terms of symmetric polynomials gives:\n[3,0,0] >= [2,1,0]\nThis result \
follows from the majorization theorem.\n \n----------\n(x - y) (x - z)   (-x \
+ y) (y - z)   (-x + z) (-y + z)\n--------------- + ---------------- + \
----------------- >= 0\n       5                  5                  5\n      \
x                  y                  z\n                           5  5  5\n\
Multiplying both sides by x  y  z  gives:\n 6  6    6  5      5  6      5  5  \
2    6    5    5  2  5\nx  y  - x  y  z - x  y  z + x  y  z  - x  y z  + x  y \
 z  + \n \n    2  5  5      6  5    6  6    5    6      5  6    6  6\n   x  y \
 z  - x y  z  + x  z  - x  y z  - x y  z  + y  z  >= 0\nExpanding and \
collecting terms of the same sign gives:\n 6  6    5  5  2    5  2  5    2  5 \
 5    6  6    6  6\nx  y  + x  y  z  + x  y  z  + x  y  z  + x  z  + y  z  >= \
\n \n   6  5      5  6      6    5      6  5    5    6      5  6\n  x  y  z + \
x  y  z + x  y z  + x y  z  + x  y z  + x y  z\nExpressing in terms of \
symmetric polynomials gives:\n[5,5,2] + [6,6,0] >= 2 [6,5,1]\nThis follows \
from the following majorizations:\n[5,5,2] + [6,6,0] >= 2 [6,5,1]\n \n\
----------\n              8    8    8\n1   1   1    x  + y  + z\n- + - + - <= \
------------\nx   y   z       3  3  3\n               x  y  z\n               \
            3  3  3\nMultiplying both sides by x  y  z  gives:\n 8    8    8  \
   3  3  2    3  2  3    2  3  3\nx  + y  + z  >= x  y  z  + x  y  z  + x  y  \
z\nExpressing in terms of symmetric polynomials gives:\n[8,0,0] >= [3,3,2]\n\
This result follows from the majorization theorem.\n \n----------\n    1      \
     1           1           1               16\n--------- + --------- + \
--------- + --------- >= -----------------\nw + x + y   w + x + z   w + y + z \
  x + y + z    3 (w + x + y + z)\nMultiplying both sides by 3 (w + x + y) (w \
+ x + z) (w + y + z) \n \n   (x + y + z) (w + x + y + z) gives:\n   4       3 \
        2  2         3      4       3         2\n3 w  + 18 w  x + 30 w  x  + \
18 w x  + 3 x  + 18 w  y + 63 w  x y + \n \n         2         3         2  2 \
          2       2  2\n   63 w x  y + 18 x  y + 30 w  y  + 63 w x y  + 30 x  \
y  + \n \n         3         3      4       3         2             2\n   18 \
w y  + 18 x y  + 3 y  + 18 w  z + 63 w  x z + 63 w x  z + \n \n       3       \
  2                         2             2\n   18 x  z + 63 w  y z + 132 w x \
y z + 63 x  y z + 63 w y  z + \n \n         2         3         2  2          \
 2       2  2\n   63 x y  z + 18 y  z + 30 w  z  + 63 w x z  + 30 x  z  + \n \
\n           2           2       2  2         3         3         3\n   63 w \
y z  + 63 x y z  + 30 y  z  + 18 w z  + 18 x z  + 18 y z  + \n \n      4      \
  3         2  2         3       3         2\n   3 z  >= 16 w  x + 32 w  x  + \
16 w x  + 16 w  y + 64 w  x y + \n \n         2         3         2  2        \
   2       2  2\n   64 w x  y + 16 x  y + 32 w  y  + 64 w x y  + 32 x  y  + \n\
 \n         3         3       3         2             2         3\n   16 w y  \
+ 16 x y  + 16 w  z + 64 w  x z + 64 w x  z + 16 x  z + \n \n       2         \
                2             2           2\n   64 w  y z + 144 w x y z + 64 \
x  y z + 64 w y  z + 64 x y  z + \n \n       3         2  2           2       \
2  2           2\n   16 y  z + 32 w  z  + 64 w x z  + 32 x  z  + 64 w y z  + \
\n \n           2       2  2         3         3         3\n   64 x y z  + 32 \
y  z  + 16 w z  + 16 x z  + 16 y z\nExpanding and collecting terms of the \
same sign gives:\n   4      3          3      4      3        3          3    \
    3\n3 w  + 2 w  x + 2 w x  + 3 x  + 2 w  y + 2 x  y + 2 w y  + 2 x y  + \n \
\n      4      3        3        3          3        3        3      4\n   3 \
y  + 2 w  z + 2 x  z + 2 y  z + 2 w z  + 2 x z  + 2 y z  + 3 z\n \n          \
2  2    2          2        2  2        2      2  2\n    >= 2 w  x  + w  x y \
+ w x  y + 2 w  y  + w x y  + 2 x  y  + \n \n    2          2      2          \
           2          2\n   w  x z + w x  z + w  y z + 12 w x y z + x  y z + \
w y  z + \n \n      2        2  2        2      2  2        2        2      2 \
 2\n   x y  z + 2 w  z  + w x z  + 2 x  z  + w y z  + x y z  + 2 y  z\n\
Expressing in terms of symmetric polynomials gives:\n2 [3,1,0,0] + [4,0,0,0] \
>= [1,1,1,1] + [2,1,1,0] + [2,2,0,0]\nThis follows from the following \
majorizations:\n[3,1,0,0] >= [2,2,0,0]\n[3,1,0,0] >= [2,1,1,0]\n[4,0,0,0] >= \
[1,1,1,1]\n \n----------\n 2    2    2    2    2    2\nx  + y    x  + z    y  \
+ z\n------- + ------- + ------- >= x + y + z\n x + y     x + z     y + z\n\
Multiplying both sides by (x + y) (x + z) (y + z) gives:\n   3        2  2    \
    3      3        2            2        3\n2 x  y + 2 x  y  + 2 x y  + 2 x  \
z + 2 x  y z + 2 x y  z + 2 y  z + \n \n      2  2          2      2  2       \
 3        3\n   2 x  z  + 2 x y z  + 2 y  z  + 2 x z  + 2 y z  >= \n \n   3   \
     2  2      3    3        2            2      3\n  x  y + 2 x  y  + x y  + \
x  z + 4 x  y z + 4 x y  z + y  z + \n \n      2  2          2      2  2      \
3      3\n   2 x  z  + 4 x y z  + 2 y  z  + x z  + y z\nExpanding and \
collecting terms of the same sign gives:\n 3        3    3      3        3    \
  3\nx  y + x y  + x  z + y  z + x z  + y z  >= \n \n     2            2      \
      2\n  2 x  y z + 2 x y  z + 2 x y z\nExpressing in terms of symmetric \
polynomials gives:\n[3,1,0] >= [2,1,1]\nThis result follows from the \
majorization theorem.\n \n----------\n 1   1   1   1   1    3    3    3    3  \
  3\n(- + - + - + - + -) (a  + b  + c  + d  + e ) >= \n a   b   c   d   e\n \n\
      2    2    2    2    2\n  5 (a  + b  + c  + d  + e )\nMultiplying both \
sides by a b c d e gives:\n 4            4            4            4    4     \
       4\na  b c d + a b  c d + a b c  d + a b c d  + a  b c e + a b  c e + \n\
 \n        4      4            4        4          3\n   a b c  e + a  b d e \
+ a b  d e + a  c d e + a  b c d e + \n \n      3          4              3   \
       4          4\n   a b  c d e + b  c d e + a b c  d e + a c  d e + b c  \
d e + \n \n          3          4          4          4              3\n   a \
b c d  e + a b d  e + a c d  e + b c d  e + a b c d e  + \n \n          4     \
     4          4          4\n   a b c e  + a b d e  + a c d e  + b c d e  >= \
\n \n     3                3                3                3\n  5 a  b c d \
e + 5 a b  c d e + 5 a b c  d e + 5 a b c d  e + \n \n              3\n   5 a \
b c d e\nExpanding and collecting terms of the same sign gives:\n 4           \
 4            4            4    4            4\na  b c d + a b  c d + a b c  \
d + a b c d  + a  b c e + a b  c e + \n \n        4      4            4       \
 4          4\n   a b c  e + a  b d e + a b  d e + a  c d e + b  c d e + \n \n\
      4          4            4          4          4\n   a c  d e + b c  d e \
+ a b d  e + a c d  e + b c d  e + \n \n          4          4          4     \
     4\n   a b c e  + a b d e  + a c d e  + b c d e  >= \n \n     3           \
     3                3                3\n  4 a  b c d e + 4 a b  c d e + 4 a \
b c  d e + 4 a b c d  e + \n \n              3\n   4 a b c d e\nExpressing in \
terms of symmetric polynomials gives:\n[4,1,1,1,0] >= [3,1,1,1,1]\nThis \
result follows from the majorization theorem.\n \n----------\n    3           \
3           3           3             16\n--------- + --------- + --------- + \
--------- >= -------------\na + b + c   a + b + d   a + c + d   b + c + d    \
a + b + c + d\nMultiplying both sides by (a + b + c) (a + b + d) (a + c + d) \
\n \n   (b + c + d) (a + b + c + d) gives:\n   4       3         2  2         \
3      4       3         2\n3 a  + 18 a  b + 30 a  b  + 18 a b  + 3 b  + 18 a \
 c + 63 a  b c + \n \n         2         3         2  2           2       2  \
2\n   63 a b  c + 18 b  c + 30 a  c  + 63 a b c  + 30 b  c  + \n \n         3 \
        3      4       3         2             2\n   18 a c  + 18 b c  + 3 c  \
+ 18 a  d + 63 a  b d + 63 a b  d + \n \n       3         2                   \
      2             2\n   18 b  d + 63 a  c d + 132 a b c d + 63 b  c d + 63 \
a c  d + \n \n         2         3         2  2           2       2  2\n   63 \
b c  d + 18 c  d + 30 a  d  + 63 a b d  + 30 b  d  + \n \n           2        \
   2       2  2         3         3         3\n   63 a c d  + 63 b c d  + 30 \
c  d  + 18 a d  + 18 b d  + 18 c d  + \n \n      4        3         2  2      \
   3       3         2\n   3 d  >= 16 a  b + 32 a  b  + 16 a b  + 16 a  c + \
64 a  b c + \n \n         2         3         2  2           2       2  2\n   \
64 a b  c + 16 b  c + 32 a  c  + 64 a b c  + 32 b  c  + \n \n         3       \
  3       3         2             2         3\n   16 a c  + 16 b c  + 16 a  d \
+ 64 a  b d + 64 a b  d + 16 b  d + \n \n       2                         2   \
          2           2\n   64 a  c d + 144 a b c d + 64 b  c d + 64 a c  d + \
64 b c  d + \n \n       3         2  2           2       2  2           2\n   \
16 c  d + 32 a  d  + 64 a b d  + 32 b  d  + 64 a c d  + \n \n           2     \
  2  2         3         3         3\n   64 b c d  + 32 c  d  + 16 a d  + 16 \
b d  + 16 c d\nExpanding and collecting terms of the same sign gives:\n   4   \
   3          3      4      3        3          3        3\n3 a  + 2 a  b + 2 \
a b  + 3 b  + 2 a  c + 2 b  c + 2 a c  + 2 b c  + \n \n      4      3        \
3        3          3        3        3      4\n   3 c  + 2 a  d + 2 b  d + 2 \
c  d + 2 a d  + 2 b d  + 2 c d  + 3 d\n \n          2  2    2          2      \
  2  2        2      2  2\n    >= 2 a  b  + a  b c + a b  c + 2 a  c  + a b c \
 + 2 b  c  + \n \n    2          2      2                     2          2\n  \
 a  b d + a b  d + a  c d + 12 a b c d + b  c d + a c  d + \n \n      2       \
 2  2        2      2  2        2        2      2  2\n   b c  d + 2 a  d  + a \
b d  + 2 b  d  + a c d  + b c d  + 2 c  d\nExpressing in terms of symmetric \
polynomials gives:\n2 [3,1,0,0] + [4,0,0,0] >= [1,1,1,1] + [2,1,1,0] + \
[2,2,0,0]\nThis follows from the following majorizations:\n[3,1,0,0] >= \
[2,2,0,0]\n[3,1,0,0] >= [2,1,1,0]\n[4,0,0,0] >= [1,1,1,1]\n \n----------\n 1  \
 1   1   1   1    3    3    3    3    3\n(- + - + - + - + -) (v  + w  + x  + \
y  + z ) >= \n v   w   x   y   z\n \n      2    2    2    2    2\n  5 (v  + w \
 + x  + y  + z )\nMultiplying both sides by v w x y z gives:\n 4            4 \
           4            4    4            4\nv  w x y + v w  x y + v w x  y + \
v w x y  + v  w x z + v w  x z + \n \n        4      4            4        4  \
        3\n   v w x  z + v  w y z + v w  y z + v  x y z + v  w x y z + \n \n  \
    3          4              3          4          4\n   v w  x y z + w  x y \
z + v w x  y z + v x  y z + w x  y z + \n \n          3          4          4 \
         4              3\n   v w x y  z + v w y  z + v x y  z + w x y  z + v \
w x y z  + \n \n          4          4          4          4\n   v w x z  + v \
w y z  + v x y z  + w x y z  >= \n \n     3                3                3 \
               3\n  5 v  w x y z + 5 v w  x y z + 5 v w x  y z + 5 v w x y  z \
+ \n \n              3\n   5 v w x y z\nExpanding and collecting terms of the \
same sign gives:\n 4            4            4            4    4            4\
\nv  w x y + v w  x y + v w x  y + v w x y  + v  w x z + v w  x z + \n \n     \
   4      4            4        4          4\n   v w x  z + v  w y z + v w  y \
z + v  x y z + w  x y z + \n \n      4          4            4          4     \
     4\n   v x  y z + w x  y z + v w y  z + v x y  z + w x y  z + \n \n       \
   4          4          4          4\n   v w x z  + v w y z  + v x y z  + w \
x y z  >= \n \n     3                3                3                3\n  4 \
v  w x y z + 4 v w  x y z + 4 v w x  y z + 4 v w x y  z + \n \n              \
3\n   4 v w x y z\nExpressing in terms of symmetric polynomials gives:\n\
[4,1,1,1,0] >= [3,1,1,1,1]\nThis result follows from the majorization \
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