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.

Question

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.

irishboy123 · Answer

.

empty · Answer

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

anonymous · Answer

@ganeshie8 can you please help me with my question when you can this is urgent!!

ganeshie8 · Answer

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 · Answer

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

ganeshie8 · Answer

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

empty · Answer

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 · Answer

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 · Answer

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 $LHS\geq RHS$