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I'm having some difficulties with proving the following task using Muirhead inequality.

$$\cfrac{x_1}{x_2+x_3+...+x_n}+\cfrac{x_2}{x_1+x_3+...+x_n}+...+\cfrac{x_n}{x_1+x_2+...+x_{n-1}}\ge\cfrac{n}{n-1}$$

It looks like a generalized version of Nesbitt's inequality, however using Muirhead to prove classic Nesbitt seemed way easier. Any help is appreciated. Thank you

EngineerInProgress
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  • Assuming that all numbers are positive and that the second inequality misses a factor $3$ in the denominator, both follow from convexity of $(1-x)^{-1}$ (in a proper domain) and Jensen’s inequality. – WimC Dec 29 '20 at 06:49
  • @WimC. Thanks for your feedback! All numbers are positive, right. The second inequality doesn't miss anything though. This is how I found it at least. Any ideas how Muirhead might help? – EngineerInProgress Dec 29 '20 at 08:01
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    In the second inequality, if all numbers are $1$ then $\tfrac43 \geq 4$? – WimC Dec 29 '20 at 09:16
  • Ah, you are right. Clearly a mistake in the task. Thanks :) – EngineerInProgress Dec 29 '20 at 09:32

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For simplicity, let's write $\sum x_i =S$ and $A_j= S-x_j$ We need to prove

$\sum \frac{x_i}{A_i} \geq \frac{n}{n-1}$

$ \iff \sum \frac{S}{A_i} \geq \frac{n^2}{n-1}$

$\iff S(\sum \frac{1}{A_i}) \geq \frac{n^2}{n-1}$

$\iff S(\frac{n^2}{\sum A_i}) \geq \frac{n^2}{n-1}$ by AM-HM

$\iff S(\frac{n^2}{(n-1)S}) \geq \frac{n^2}{n-1}$.

Mathematical Curiosity
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