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let $x,y,z$ are real numbers,How prove this inequality $$\sum_{cyc}\left(\dfrac{x^3}{(y+z)^3}-\dfrac{y^3z^3}{(x^2+yz)^3}\right)\ge 0$$

My idea want use the SOS methods, but it is very ugly, is there any nicer method? Thank you

Example problems using the SOS method at: http://ineqkhoinguyen.files.wordpress.com/2008/02/ren-luyen-sos.pdf

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  • $\begingroup$ Can you explain what $\sum_{cyc}$ means? $\endgroup$ – Thomas Andrews Jul 18 '13 at 15:59
  • $\begingroup$ such $\sum_{cyc}a=a+b+c,\sum_{cyc}a^2b=a^2b+b^2c+c^2a,\cdots$ $\endgroup$ – math110 Jul 18 '13 at 16:02
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    $\begingroup$ It's not clear why that link is there, since it doesn't clarify what the SOS method is. $\endgroup$ – Thomas Andrews Jul 18 '13 at 16:55
  • $\begingroup$ I know only two articles in English that describe SOS (sum of squares), here is one by David Arthur and here is one by Pham Kim Hung (the second link is direct link to PDF, the first isn't). $\endgroup$ – nikoma Jul 21 '13 at 20:57
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It's wrong! Try $y+z=0$.

For positive variables your inequality is true but not so easy (at lest my proof is very ugly)

and I think SOS does not help here.

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