Inequality with a,b,c, positive numbers, a+b+c = 1

Question

$a,b,c$ are positive real numbers and $a+b+c = 1$. Prove that $$ \frac{1}{1-a} + \frac{1}{1-b} + \frac{1}{1-c} \ge \frac{2}{1+a} + \frac{2}{1+b} + \frac{2}{1+c} $$

I have tried using cauchy schwarz on both sides, but that just gives me LHS $\ge$ 4.5 and RHS $ \ge$ 4.5. I also tried bashing it out but it turns out very ugly. I don't think I'm utilizing the $a+b+c= 1$ part enough, but I'm kind of stuck on this problem.

Help is appreciated.

Answer 1

Hint: by AM-HM inequality$$\frac{1}{1-a}+\frac{1}{1-b}\ge \frac{1}{b+c}+\frac{1}{a+c}\ge \frac{4}{a+b+2c}$$