# Strange Mean Inequality [duplicate]

This problem was inspired by this question.

$\sqrt [ 3 ]{ a(\frac { a+b }{ 2 } )(\frac { a+b+c }{ 3 } ) } \ge \frac { a+\sqrt { ab } +\sqrt [ 3 ]{ abc } }{ 3 }$

The above can be proved using Hölder's inequality.

$\sqrt [ 3 ]{ a(\frac { a+b }{ 2 } )(\frac { a+b+c }{ 3 } ) } =\sqrt [ 3 ]{ (\frac { a }{ 3 } +\frac { a }{ 3 } +\frac { a }{ 3 } )(\frac { a }{ 3 } +\frac { a+b }{ 6 } +\frac { b }{ 3 } )(\frac { a+b+c }{ 3 } ) } \ge \sqrt [ 3 ]{ (\frac { a }{ 3 } +\frac { a }{ 3 } +\frac { a }{ 3 } )(\frac { a }{ 3 } +\frac { \sqrt { ab } }{ 3 } +\frac { b }{ 3 } )(\frac { a }{ 3 } +\frac { b }{ 3 } +\frac { c }{ 3 } ) } (\because \text{AM-GM})\\ \ge \frac { a+\sqrt { ab } +\sqrt [ 3 ]{ abc } }{ 3 } (\because \text{Holder's inequality)}$

However, I had trouble generalizing this inequality to

$\sqrt [ n ]{ \prod _{ i=1 }^{ n }{ { A }_{ i } } } \ge \frac { \sum _{ i=1 }^{ n }{ { G }_{ i } } }{ n }$

when ${ A }_{ i }=\frac { \sum _{ j=1 }^{ i }{ { a }_{ i } } }{ i }$

and ${ G }_{ i }=\sqrt [ i ]{ \prod _{ j=1 }^{ i }{ { a }_{ i } } }$ as I could not split the fractions as I did above.

• Do you have reason to believe that the generalization is true? Jan 3, 2016 at 4:51
• Yes, I do. I did manage to formulate a proof for n=4,5,6. However, the way I proved them is the same as n=3, and I believed that it was not necesarry to post them. However, I had trouble repeating the process for n>6. So it could be wrong. If you do find a counter example, please inform me. Jan 3, 2016 at 4:58
• Well, the generalization of this problem is true. But the proof is difficult. :) Jan 3, 2016 at 5:31
• @貓貓吃狗狗 I do not mean to annoy you, but am I correct in my assumption that your comment indicated your intent to give a solution/hint to the problem? Jan 5, 2016 at 6:28
• This is a result of K. Kedlaya, "K. KEDLAYA, Proof of a mixed arithmetic-mean, geometric-mean inequality, Amer. Math. Monthly, 101 (1994), 355–357." A generalization could be found in the article : emis.de/journals/JIPAM/images/165_06_JIPAM/165_06.pdf Jan 9, 2016 at 19:43