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Question Number 143139 by loveineq last updated on 10/Jun/21

Let a,b,c > 0 and a+b+c = 3. Prove that                   (1+a^2 )(1+b^2 )(1+c^2 ) ≤ (1+(1/( (√(abc)))))^3

$$\mathrm{Let}\:{a},{b},{c}\:>\:\mathrm{0}\:\mathrm{and}\:{a}+{b}+{c}\:=\:\mathrm{3}.\:\mathrm{Prove}\:\mathrm{that} \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{1}+{a}^{\mathrm{2}} \right)\left(\mathrm{1}+{b}^{\mathrm{2}} \right)\left(\mathrm{1}+{c}^{\mathrm{2}} \right)\:\leqslant\:\left(\mathrm{1}+\frac{\mathrm{1}}{\:\sqrt{{abc}}}\right)^{\mathrm{3}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$ $$ \\ $$

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