if-a-b-c-lt-0-and-a-b-c-3-prove-that-1-b-a-1-b-2-1-c-b-1-c-2-1-a-c-1-a-2-8-

Question Number 164416 by HongKing last updated on 16/Jan/22

if a;b;c<0 and a+b+c=3 prove that: (1 + (b/a))^(1/b^2 ) ∙ (1 + (c/b))^(1/c^2 ) ∙ (1 + (a/c))^(1/a^2 ) ≥ 8

$$\mathrm{if}\:\:\mathrm{a};\mathrm{b};\mathrm{c}<\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{3}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\left(\mathrm{1}\:+\:\frac{\mathrm{b}}{\mathrm{a}}\right)^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{b}}^{\mathrm{2}} }} \centerdot\:\left(\mathrm{1}\:+\:\frac{\mathrm{c}}{\mathrm{b}}\right)^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{c}}^{\mathrm{2}} }} \centerdot\:\left(\mathrm{1}\:+\:\frac{\mathrm{a}}{\mathrm{c}}\right)^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{a}}^{\mathrm{2}} }} \geqslant\:\mathrm{8} \\ $$

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