Let $x,y,z>0$ and such $xyz\ge 1$,show that $$\sum_{cyc}\dfrac{2}{2+x+3y}\le\sum_{cyc}\dfrac{1}{2+x}$$I tried C-S,Jenson inequality but without success.
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It might be clearer to expand your sums to include all terms. Since $z$ does not appear in either summand expression, and $y$ does not appear in the RHS, it's unclear what a "cyclic" summation might mean. – hardmath Jul 23 '17 at 23:54
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@wightahtl I have a proof for $xyz=1$. If you wish I can show. – Michael Rozenberg Jul 24 '17 at 04:59
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I see it, Thanks – wightahtl Jul 24 '17 at 05:10