## zmudz one year ago 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.

1. IrishBoy123

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2. Empty

Well two simple cases where equality hold: $$x_i=1$$ for all i or n=1

3. anonymous

4. ganeshie8

I think we may use this inequality http://www.artofproblemsolving.com/wiki/index.php/Muirhead's_Inequality notice that $$(93,19,0,\ldots) \succ (92,20,0,\ldots)$$

5. ikram002p

thinking of what @Empty have said i'll add its equal whenever xi a constant for all i :D

6. ganeshie8

wiki says that muirhead is actually a generalization of Jensen's inequality https://en.wikipedia.org/wiki/Jensen%27s_inequality

7. 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

8. 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.

9. 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 $$LHS\geq RHS$$