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$a,b,c \geqslant 0$ and $a+b+c=3$, prove that

$$ \sum_{cyc} \left( \frac{ab}{1+b^3}\right)^ \frac54 \leqslant 3 \cdot \left(\frac12\right)^\frac54 $$

I did not know how to handle the exponential term $n=\frac54$ in this inequality. I saw a similar problem on AOPS, which is posted here. I tried the similar approach but fail.

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    This seems easy enough to brute force with Lagrange Multipliers or the like, since we only have two variables. – 2017-03-10
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    Any luck with Lagrange yet? – 2017-03-11
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    No it is too much computation for Lagrange – 2017-03-11
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    @BrevanEllefsen there's three variables – 2017-03-12
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    @vrugtehagel not sure what you meant by that. Lagrange multipliers work with any number of variables... You must simply solve $n-1$ equations simultaneously, which gets a lot harder as more variables are added – 2017-03-14
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    "..since we only have two variables." I was merely correcting you from your first comment. I made no statement about Lagrange multipliers – 2017-03-14

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