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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.

Chad Shin
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  • Do you have reason to believe that the generalization is true? – David Jan 03 '16 at 04:51
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    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. – Chad Shin Jan 03 '16 at 04:58
  • Well, the generalization of this problem is true. But the proof is difficult. :) – 貓貓吃狗狗 Jan 03 '16 at 05:31
  • @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? – Chad Shin Jan 05 '16 at 06:28
  • 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 : http://www.emis.de/journals/JIPAM/images/165_06_JIPAM/165_06.pdf – Omran Kouba Jan 09 '16 at 19:43
  • @OmranKouba Here is a link to Kedlaya's article. http://www.dma.ulpgc.es/profesores/personal/aph/ficheros/resolver/ficheros/crp/mixed_1994.pdf Please write this as an answer and I will give you the bounty. – David Jan 09 '16 at 21:08

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This result was conjectured by professor Finbarr Holand, and then it was proved by K. Kedlaya in an article in the American Mathematical Monthly that could be found here, and then it was generalized by Professor Holand in an article that could be found here.

Omran Kouba
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