zmudz
  • zmudz
Prove that if \(x_i > 0\) for all \(i\) then \begin{align*} &(x_1^{19} + x_2^{19} + \cdots + x_n^{19})(x_1^{93} + x_2^{93} + \cdots + x_n^{93}) \\ &\geq (x_1^{20} + x_2^{20} + \cdots + x_n^{20})(x_1^{92} + x_2^{92} + \cdots + x_n^{92}). \end{align*} Also, find when equality holds.
Mathematics
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SOLVED
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katieb
  • katieb
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IrishBoy123
  • IrishBoy123
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Empty
  • Empty
Well two simple cases where equality hold: \(x_i=1\) for all i or n=1
anonymous
  • anonymous
@ganeshie8 can you please help me with my question when you can this is urgent!!

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ganeshie8
  • ganeshie8
I think we may use this inequality http://www.artofproblemsolving.com/wiki/index.php/Muirhead's_Inequality notice that \((93,19,0,\ldots) \succ (92,20,0,\ldots) \)
ikram002p
  • ikram002p
thinking of what @Empty have said i'll add its equal whenever xi a constant for all i :D
ganeshie8
  • ganeshie8
wiki says that muirhead is actually a generalization of Jensen's inequality https://en.wikipedia.org/wiki/Jensen%27s_inequality
Empty
  • Empty
I was actually looking at the Caucy-Schwartz inequality but I don't know how to solve this problem yet I am trying to cook up some ideas though. https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality
zmudz
  • zmudz
Thanks! I know the Cauchy-Schwarz inequality from class; however, we never covered the Jensen's inequality or Muirhead. But any help at all is much appreciated. Let me know how it goes. I'm still stuck.
Loser66
  • Loser66
WOLG \(x_1\leq x_2\leq x_3\leq.....\leq x_n\) LHS:= \[x_1^{19}\left[\begin{matrix}x_1^{93}\\x_2^{93}\\::::\\x_n^{93}\end{matrix}\right]+x_2^{19}\left[\begin{matrix}x_1^{93}\\x_2^{93}\\::::\\x_n^{93}\end{matrix}\right]+::::+x_n^{19}\left[\begin{matrix}x_1^{93}\\x_2^{93}\\::::\\x_n^{93}\end{matrix}\right]\] RHS:= \[x_1^{20}\left[\begin{matrix}x_1^{92}\\x_2^{92}\\::::\\x_n^{92}\end{matrix}\right]+x_2^{20}\left[\begin{matrix}x_1^{92}\\x_2^{92}\\::::\\x_n^{92}\end{matrix}\right]+::::+x_n^{20}\left[\begin{matrix}x_1^{92}\\x_2^{92}\\::::\\x_n^{92}\end{matrix}\right]\] The first entry of the first matrix of the LHS is \(x_1^{19}*x_1^{93}=x_1^{112}\geq x_1^{20}*x_1^{92}=x_1^{112}\) og the RHS The second entry: LHS \(x_1^{19}*x_2^{93}=x_1^{19}*x_2^{92}x_2\geq x_1^{19}*x_2^{92}x_1=x_1^{20}*x_2^{92}=\text{the second entry of the RHS}\) Same as other entries and other matrix. That shows \(LHS\geq RHS\)

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