How prove this inequality ∑ c y c a 4 a 2 + 2 b 4 ≥ 1Let a , b , c &gt; 0 such a + b + c = 3. Show that a 4 a 2 + 2 b 4 + b 4 b 2 + 2 c 4 + c 4 c 2 + 2 a 4 ≥ 1My attempt is to use Cauchy-Schwarz inequality. Hence, I consider ( a 4 a 2 + 2 b 4 + b 4 b 2 + 2 c 4 + c 4 c 2 + 2 a 4 ) ( a 2 + b 2 + c 2 + 2 a 4 + 2 b 4 + 2 c 4 ) ≥ ( a 2 + b 2 + c 2 ) 2 However, ( a 2 + b 2 + c 2 ) 2 ≤ ( a 2 + b 2 + c 2 + 2 a 4 + 2 b 4 + 2 c 4 )

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How prove this inequality       ∑          c      y      c                  a      4                      a        2            +      2              b        4              ≥  1Let   a  ,  b  ,  c  &amp;gt;  0 such   a  +  b  +  c  =  3. Show that                    a        4                              a          2                +        2                  b          4                      +                    b        4                              b          2                +        2                  c          4                      +                    c        4                              c          2                +        2                  a          4                      ≥  1My attempt is to use Cauchy-Schwarz inequality. Hence, I consider      (                            a          4                                      a            2                    +          2                      b            4                                +                            b          4                                      b            2                    +          2                      c            4                                +                            c          4                                      c            2                    +          2                      a            4                                )    (      a    2    +      b    2    +      c    2    +  2      a    4    +  2      b    4    +  2      c    4    )  ≥  (      a    2    +      b    2    +      c    2        )    2  However,  (      a    2    +      b    2    +      c    2        )    2    ≤  (      a    2    +      b    2    +      c    2    +  2      a    4    +  2      b    4    +  2      c    4    )

optativaspv · Accepted Answer

By C-S      ∑          c      y      c                  a      4                      a        2            +      2              b        4              =      ∑          c      y      c                  a      8                      a        6            +      2              a        4                    b        4              ≥            (              a        4            +              b        4            +              c        4                    )        2                            ∑                  c          y          c                    (              a        6            +      2              a        4                    b        4            )        .Thus, it remains to prove that  (      a    4    +      b    4    +      c    4        )    2    ≥      ∑          c      y      c        (      a    6    +  2      a    4        b    4    )or      a    8    +      b    8    +      c    8    ≥      a    6    +      b    6    +      c    6  or  9  (      a    8    +      b    8    +      c    8    )  ≥  (  a  +  b  +  c      )    2    (      a    6    +      b    6    +      c    6    )  ,which is obvious by Power Mean inequality:            (                                    a            8                    +                      b            8                    +                      c            8                          3            )                      2        8              ≥            (                        a          +          b          +          c                3            )        2  and                    a        6            +              b        6            +              c        6              3    ≤            (                                    a            8                    +                      b            8                    +                      c            8                          3            )                      6        8              .Done!The inequality       a    8    +      b    8    +      c    8    ≥      a    6    +      b    6    +      c    6   we can prove also by the following way.We need to prove that      ∑          c      y      c        (      a    8    −      a    6    )  ≥  0or      ∑          c      y      c        (      a    6    (  a  −  1  )  (  a  +  1  )  −  2  (  a  −  1  )  )  ≥  0or      ∑          c      y      c        (  a  −  1  )  (      a    7    −  1  +      a    6    −  1  )  ≥  0  ,which is obvious.

Let a,b,c>0 such a+b+c=3. Show that (a^4)/(a^2+2b^4)+(b^4)/(b^2+2c^4)+(c^4)/(c^2+2a^4)>= 1

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