inequality with three variables and conditionIf a, b and c positive real numbers such that a + b + c = 1, prove b 2 a + b 2 + c 2 b + c 2 + a 2 c + a 2 ⩾ 3 4 . I have tried several methods to solve this,but can't get any result. Any idea?

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inequality with three variables and conditionIf   a,   b and   c positive real numbers such that   a  +  b  +  c  =  1, prove                     b        2                    a        +                  b          2                      +                    c        2                    b        +                  c          2                      +                    a        2                    c        +                  a          2                      ⩾            3      4      . I have tried several methods to solve this,but can&#039;t get any result. Any idea?

laure6237ma · Accepted Answer

By C-S and Vasc we obtain       ∑          c      y      c                  a      2                      a        2            +      c        =      ∑          c      y      c                  a      4                      a        4            +              a        2            c      (      a      +      b      +      c      )        ≥            (              a        2            +              b        2            +              c        2                    )        2                            ∑                  c          y          c                    (              a        4            +              a        3            c      +              a        2                    b        2            +              a        2            b      c      )        ≥  ≥            (              a        2            +              b        2            +              c        2                    )        2                            ∑                  c          y          c                    (              a        4            +              a        2                    b        2            +              a        2            b      c      )      +                        (                      a            2                    +                      b            2                    +                      c            2                                )            2                          3              =            3      (              a        2            +              b        2            +              c        2                    )        2                            ∑                  c          y          c                    (      4              a        4            +      5              a        2                    b        2            +      3              a        2            b      c      )      Id est, it remains to prove that   4  (      a    2    +      b    2    +      c    2        )    2    ≥      ∑          c      y      c        (  4      a    4    +  5      a    2        b    2    +  3      a    2    b  c  ), which is       ∑          c      y      c            c    2    (  a  −  b      )    2    ≥  0Done!

inequality with three variables and condition If a , b and c positive real numbers

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