Inequality - GM, AM, HM and SM meansI've got stuck at this problem :Prove that for any a &gt; 0 and any b &gt; 0 the following inequality is true: 3 ( a 3 b 3 + b 3 a 3 ) ≥ a b + b a + 4The first thing that I've thought was the AM-GM inequality (the extended version - heard that is also known as The power mean inequality): H M ≤ G M ≤ A M ≤ S Mwhere H M, G M, A M, and S M refer to the harmonic, geometric, arithmetic, and square mean, respectively. CBS(Cauchy - Buniakowsky - Schwartz) also come to my mind, but I think it isn't helpful in this case.I would be greatful for some hints.Thanks!

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Inequality - GM, AM, HM and SM meansI&#039;ve got stuck at this problem :Prove that for any   a  &amp;gt;  0 and any   b  &amp;gt;  0 the following inequality is true:      3              (                        a          3                          b          3                    +                        b          3                          a          3                    )        ≥      a    b    +      b    a    +  4The first thing that I&#039;ve thought was the AM-GM inequality (the extended version - heard that is also known as The power mean inequality):  H  M  ≤  G  M  ≤  A  M  ≤  S  Mwhere   H  M,   G  M,   A  M, and   S  M refer to the harmonic, geometric, arithmetic, and square mean, respectively. CBS(Cauchy - Buniakowsky - Schwartz) also come to my mind, but I think it isn&#039;t helpful in this case.I would be greatful for some hints.Thanks!

Liberty Gates · Accepted Answer

Let   x  =      a    b    +      b    a    ≥  2 (by AM-GM). Then      x    3    =            a      3              b      3        +            b      3              a      3        +  3  xSo we want to show, for   x  ≥  2  3  (      x    3    −  3  x  )  ≥  x  +  4    ⟺    (  x  −  2  )  (  3      x    2    +  6  x  +  2  )  ≥  0which is obvious.

Jude Hunt · Accepted Answer

this inequation is equivalent to            (      a      −      b              )        2                    (        3                  a          4                +        6                  a          3                b        +        8                  a          2                          b          2                +        6        a                  b          3                +        3                  b          4                )                            a        3                    b        3              ≥  0which is true.

Inequality - GM, AM, HM and SM means I've got stuck at this problem : Prove that for any a

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