a 3 b 2 + b 3 c 2 + c 3 a 2 ≥ 3 a 2 + b 2 + c 2 a + b + c PropositionFor any positive numbers a, b, and c,I am requesting an elementary, algebraic explanation to this inequality. (I suppose the condition for equality is that a=b=c.) I am not familiar with symmetric inequalities in three variables. I would appreciate any references.

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a      3              b      2        +            b      3              c      2        +            c      3              a      2        ≥  3                      a        2            +              b        2            +              c        2                    a      +      b      +      c      PropositionFor any positive numbers a, b, and c,I am requesting an elementary, algebraic explanation to this inequality. (I suppose the condition for equality is that a=b=c.) I am not familiar with symmetric inequalities in three variables. I would appreciate any references.

Miya Swanson · Accepted Answer

By C-S and Holder we obtain:      ∑          c      y      c                  a      3              b      2        =      ∑          c      y      c                  a      5                      a        2                    b        2              ≥                    (                  a                                    5              2                                      +                  a                                    5              2                                      +                  a                                    5              2                                      )            2                      ∑                  c          y          c                            a        2                    b        2              =  =                              (                      a                                          5                2                                              +                      a                                          5                2                                              +                      a                                          5                2                                              )                2            (      a      +      b      +      c      )              (      a      +      b      +      c      )              ∑                  c          y          c                            a        2                    b        2              ≥            (              a        2            +              b        2            +              c        2                    )        3                    (              a        2                    b        2            +              a        2                    c        2            +              b        2                    c        2            )      (      a      +      b      +      c      )        ≥  ≥            3      (              a        2                    b        2            +              a        2                    c        2            +              b        2                    c        2            )      (              a        2            +              b        2            +              c        2            )              (              a        2                    b        2            +              a        2                    c        2            +              b        2                    c        2            )      (      a      +      b      +      c      )        =            3      (              a        2            +              b        2            +              c        2            )              a      +      b      +      c        .Done!

Proposition For any positive numbers a, b, and c, (a^3)/(b^2) + (b^3)/(c^2) + (c^3)/(a^2) >= 3 (a^2 + b^2 + c^2)/(a + b + c)

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