In AoPS' (Art of Problem Solving) proof of Muirhead's inequality, how does the below equality work out?The below equation appears to show two expressions (1 and 2), each under the symmetric sum notation, being multiplied together to produce expression 3, which is also under the symmetric sum notation...but normally, [ A 1 + A 2 ( … ) + A n ] ⋅ [ B 1 + B 2 + ( … ) + B n ] is not [ A 1 B 1 + A 2 B 2 + ( … ) + A n ⋅ B n ] ,that is, when we multiply the sum of A i ( i = [ 1 , n ] ) and B i ( i = [ 1 , n ] ), the result is not the sum of A i ⋅ B i ( i = [ 1 , n ] )

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In AoPS&#039; (Art of Problem Solving) proof of Muirhead&#039;s inequality, how does the below equality work out?The below equation appears to show two expressions (1 and 2), each under the symmetric sum notation, being multiplied together to produce expression 3, which is also under the symmetric sum notation...but normally,  [      A          1        +      A          2        (  …  )  +      A          n        ]  ⋅  [      B          1        +      B          2        +  (  …  )  +      B          n        ] is not   [      A    1        B    1    +      A    2        B    2    +  (  …  )  +      A    n    ⋅      B    n    ] ,that is, when we multiply the sum of       A    i    (  i  =  [  1  ,     n  ]  ) and       B    i    (  i  =  [  1  ,     n  ]  ), the result is not the sum of       A    i    ⋅      B    i    (  i  =  [  1  ,     n  ]  )

Punktatsp · Accepted Answer

Step 1Yes, that proof is bogus. If you examine it, they only use the fact that   ∑      a    i    =  ∑      b    i   and not the other conditions of majorization. So this &quot;proof&quot; purports to show that if   ∑      a    i    =  ∑      b    i   then      ∑          σ      ∈              S        n                  x          σ      (      1      )                      a        1              ⋯      x          σ      (      n      )                      a        n              ≥      ∑          σ      ∈              S        n                  x          σ      (      1      )                      b        1              ⋯      x          σ      (      n      )                      b        n              . .(And hence they are equal, as well.) I mean this just can&#039;t be true for trivial reasons.

In AoPS' (Art of Problem Solving) proof of Muirhead's inequality, how does the below equality work o

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