If a,b,c ∈R^+ with a+b+c=3 prove that (1/(a^6 +b^6 +3c^3 +4))+(1/(b^6 +c^6 +3a^3 +4))+(1/(c^6 +a^6 +3b^3 +4))≤(3/(3+2((√(ab))+(√(bc))+(√(ac)))))


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Question Number 203379 by York12 last updated on 18/Jan/24

$If a, b, c \in R^{+} with a + b + c = 3 prove that$ $\frac{1}{a^{6} + b^{6} + 3 c^{3} + 4} + \frac{1}{b^{6} + c^{6} + 3 a^{3} + 4} + \frac{1}{c^{6} + a^{6} + 3 b^{3} + 4} ⩽ \frac{3}{3 + 2 (\sqrt{a b} + \sqrt{b c} + \sqrt{a c})}$
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