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Question Number 78340 by loveineq. last updated on 16/Jan/20

Let a, b, c > 0 and c^{2} = \frac{a b + b c + c a}{3} . Prove that

\frac{a^{3} + b^{3} - 2 c^{3}}{a^{3} + b^{3} + c^{3}} ⩽ 3 (\frac{a^{2} + b^{2} - 2 c^{2}}{a^{2} + b^{2} + c^{2}})

Answered by MJS last updated on 16/Jan/20

(a^{3} + b^{3} - 2 c^{3}) (a^{2} + b^{2} + c^{2}) ⩽ 3 (a^{2} + b^{2} - 2 c^{2}) (a^{3} + b^{3} + c^{3})

\Rightarrow

2 (a^{5} + b^{5}) - 7 c^{2} (a^{3} + b^{3}) + 5 c^{3} (a^{2} + b^{2}) + 2 a^{2} b^{2} (a + b) - 4 c^{5} ⩾ 0

let a = p c \land b = q c

(2 (p^{5} + q^{5}) + 2 p^{2} q^{2} (p + q) - 7 (p^{3} + q^{3}) + 5 (p^{2} + q^{2}) - 4) c^{5} ⩾ 0

2 (p^{5} + q^{5}) + 2 p^{2} q^{2} (p + q) - 7 (p^{3} + q^{3}) + 5 (p^{2} + q^{2}) - 4 ⩾ 0

3 c^{2} = a b + a c + b c

3 c^{2} = (p + p q + q) c^{2} \Rightarrow q = \frac{3 - p}{1 + p}

\frac{2 p^{10} + 10 p^{9} + 15 p^{8} - 18 p^{7} - 33 p^{6} - 12 p^{5} - 33 p^{4} - 90 p^{3} + 735 p^{2} - 914 p + 338}{{(p + 1)}^{5}} ⩾ 0

{(p - 1)}^{2} (2 p^{8} + 14 p^{7} + 41 p^{6} + 50 p^{5} + 26 p^{4} - 10 p^{3} - 79 p^{2} - 238 p + 338) ⩾ 0

no other real linear factors

with p = 1 lhs = 0

with p <> 1 lhs > 0

\Rightarrow proven

Answered by loveineq. last updated on 16/Jan/20

Thanks MJS