Prove-or-disprove-2ab-2bc-ca-5-3-2-abc-a-b-c-gt-0-

Question Number 6154 by Rasheed Soomro last updated on 16/Jun/16

P r o v e o r d i s p r o v e

{(\frac{2 a b + 2 b c + c a}{5})}^{\frac{3}{2}} ⩾ a b c \forall a, b, c > 0

Commented by Yozzii last updated on 18/Jun/16

\frac{5 \sqrt{5}}{(\frac{2}{c} + \frac{2}{a} + \frac{1}{b}) \sqrt{2 a b + 2 b c + c a}} ⩽ 1

\frac{1}{b} + \frac{2}{a} + \frac{2}{c} ⩾ 5 \sqrt{\frac{5}{2 a b + 2 b c + c a}}

{(\frac{1}{b} + \frac{2}{a} + \frac{2}{c})}^{2} ⩾ \frac{125}{2 a b + 2 b c + c a}

{(\frac{1}{b} + \frac{2}{a} + \frac{2}{c})}^{3} ⩾ \frac{125}{a b c}

\frac{1}{3} (\frac{1}{b} + \frac{2}{a} + \frac{2}{c}) ⩾ \frac{1}{3} {(\frac{125}{a b c})}^{1 / 3}

\frac{1}{3} (\frac{1}{b} + \frac{2}{a} + \frac{2}{c}) ⩾ (1.666 \dots) . \frac{1}{{(a b c)}^{1 / 3}}

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B y A M - G M,

\frac{b^{- 1} + 2 a^{- 1} + 2 c^{- 1}}{3} ⩾ {(b^{- 1} \times 2 a^{- 1} \times 2 c^{- 1})}^{1 / 3}

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