Elementary Symmetric Means as Quasi-Arithmetic MeansThe elementary symmetric polynomial of degree k in n variables is e n , k ( x ) = ∑ 1 ≤ i 1 ≤ i 2 ≤ ⋯ ≤ i k ≤ n x i 1 x i 2 ⋯ x i k and for positive data x, the corresponding elementary symmetric mean is s n , k ( x ) := e n , k ( x ) ( n k ) k .Quasi-arithmetic means are defined in terms of an invertible function f, and the arithmetic mean A M f ( x ) := ( f − 1 ∘ A ∘ f ) ( x ) = f − 1 ( f ( x 1 ) + f ( x 2 ) + ⋯ + f ( x n ) n ) .For any n, the elementary symmetric mean in n variables of degree n is the geometric mean, which is a quasi-arithmetic mean: s n , n ( x 1 , … , x n ) = x 1 x 2 ⋯ x n n = exp ⁡ ( log ⁡ ( x 1 ) + log ⁡ ( x 2 ) + ⋯ + log ⁡ ( x n ) n ) .My question is: are the other elementary symmetric means (with k &lt; n) also quasi-arithmetic? If so, can the conjugating functions f n , k be described explicitly?My attempt: Obviously s n , 1 = A is quasi-arithmetic, so the first nontrivial case is s 3 , 2 , with the functional equation f 3 , 2 ∘ s 3 , 2 = A ∘ f 3 , 2 ,or f 3 , 2 ( x 1 x 2 + x 1 x 3 + x 2 x 3 3 ) = f 3 , 2 ( x 1 ) + f 3 , 2 ( x 2 ) + f 3 , 2 ( x 3 ) 3 .I tried pre-composing f with a logarithm and differentiation, but it didn't seem to get me anywhere.I would greatly appreciate any help on this.

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Elementary Symmetric Means as Quasi-Arithmetic MeansThe elementary symmetric polynomial of degree k in n variables is      e          n      ,      k        (      x    )  =      ∑          1      ≤              i        1            ≤              i        2            ≤      ⋯      ≤              i        k            ≤      n            x                  i        1                  x                  i        2              ⋯      x                  i        k            and for positive data x, the corresponding elementary symmetric mean is      s          n      ,      k        (      x    )  :=                              e                      n            ,            k                          (                  x                )                                          (                                    n          k                                      )                                k    .Quasi-arithmetic means are defined in terms of an invertible function f, and the arithmetic mean A      M    f    (      x    )  :=  (      f          −      1        ∘  A  ∘  f  )  (      x    )  =      f          −      1            (                  f        (                  x          1                )        +        f        (                  x          2                )        +        ⋯        +        f        (                  x          n                )            n        )    .For any n, the elementary symmetric mean in n variables of degree n is the geometric mean, which is a quasi-arithmetic mean:      s          n      ,      n        (      x    1    ,  …  ,      x    n    )  =                    x        1                    x        2            ⋯              x        n              n    =  exp  ⁡      (                  log        ⁡        (                  x          1                )        +        log        ⁡        (                  x          2                )        +        ⋯        +        log        ⁡        (                  x          n                )            n        )    .My question is: are the other elementary symmetric means (with   k  &amp;lt;  n) also quasi-arithmetic? If so, can the conjugating functions       f          n      ,      k       be described explicitly?My attempt: Obviously       s          n      ,      1        =  A is quasi-arithmetic, so the first nontrivial case is       s          3      ,      2      , with the functional equation      f          3      ,      2        ∘      s          3      ,      2        =  A  ∘      f          3      ,      2        ,or      f          3      ,      2            (                                        x            1                                x            2                    +                      x            1                                x            3                    +                      x            2                                x            3                          3              )    =                    f                  3          ,          2                    (              x        1            )      +              f                  3          ,          2                    (              x        2            )      +              f                  3          ,          2                    (              x        3            )        3    .I tried pre-composing f with a logarithm and differentiation, but it didn&#039;t seem to get me anywhere.I would greatly appreciate any help on this.

Davin Meyer · Accepted Answer

Step 1If you require that f is, say, C1, then differentiating Mf with respect to any xi gives      ∂          ∂              x        i                  M    f    =      1                  f        ′                    (                              ∑            f            (                          x              i                        )                    n                )                                f        ′            (              x        i            )        n  which means that for any   i  ≠  j we have                    ∂                  ∂                      x            i                                      M        f                            ∂                  ∂                      x            j                                      M        f              =                    f        ′            (              x        i            )                      f        ′            (              x        j            )        .So we can check whether       s          3      ,      2       has this property. Actually it will be slightly more convenient for the purposes of calculating derivatives to check this property after conjugating       s          3      ,      2       by   f  (  x  )  =      x    2   to remove the outer square root, giving a modified mean      t          3      ,      2        (      x    1    ,      x    2    ,      x    3    )  =                              x          1                          x          2                    +                        x          2                          x          3                    +                        x          3                          x          1                      3    .Step 2(Conjugating preserves the property of being quasi-arithmetic so this is fine.) We get      ∂          ∂              x        1                  t          3      ,      2        =                              x          2                    +                        x          3                            6                        x          1                    and similarly for       x    2    ,      x    3  , which gives                    ∂                  ∂                      x            1                                      t                  3          ,          2                                    ∂                  ∂                      x            2                                      t                  3          ,          2                      =            (                        x          2                    +                        x          3                    )                        x          2                            (                        x          1                    +                        x          3                    )                        x          1                      .In particular, this quotient depends nontrivially on       x    3  , so it&#039;s not of the form                     f        ′            (              x        1            )                      f        ′            (              x        2            )       for any function f. A similar but more annoying calculation can be done for the other elementary symmetric means. Intuitively this is saying that the elementary symmetric means &quot;mix&quot; the       x    i   too much to be quasi-arithmetic.

Elementary Symmetric Means as Quasi-Arithmetic Means. The elementary symmetric polynomial of degree k in n variables is e_{n,k}(x)=sum_{1 <= i_1 <= i_2 <= cdots <= i_k <= n}x_{i_1}x_{i_2} cdots x_{i_k}

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