Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality $$\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$$?
Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality $\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$?
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5Hint: Observe that it's sufficient to prove the inequality $\frac{\sqrt{x}}{1-x}-\frac{3\sqrt{3}}{2}x \geq 0$, when $x\in [0,1)$. Can you prove this one ? – 2012-08-10
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0@RaduTitiu thank you.the constraint $a+b+c=1$ can be weaken by $0 – 2012-08-10
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0and if I want to proof $$\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}+\frac{\sqrt{d}}{1-d} \geq \frac{8}{3}$$for $a+b+c+d=1$?what's more,$$\sum_{i=1}^n\frac{\sqrt{a_i}}{1-a_i} \geq \frac{n^{\frac{3}{2}}}{n-1}$$for $\sum_{i=1}^n a_i=1$ – 2012-08-10
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0@RaduTitiu Yep. – 2012-08-10