$a+b+c=1.$ Show that $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c} \geq \frac{2}{1+a}+\frac{2}{1+b}+\frac{2}{1+c}$

Question

Assume $a,b,c>0$ and $a+b+c=1.$ Show that $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c} \geq \frac{2}{1+a}+\frac{2}{1+b}+\frac{2}{1+c}$

Here's what I tried:

$\frac{a+b+c}{b+c}+\frac{a+b+c}{a+b}+\frac{a+b+c}{a+b} \geq \frac{1+a+b+c}{1+a}+\frac{1+a+b+c}{1+b}+\frac{1+a+b+c}{1+c}$ $\rightarrow \frac{a}{b+c}+\frac{c}{a+b}+\frac{b}{a+b} \geq \frac{b+c}{1+a}+\frac{a+c}{1+b}+\frac{a+b}{1+c}$

Let $A=b+c, B=a+c,$ and $ C=a+b,$ so $A+B+C=2.$

$\rightarrow \frac{1-A}{A}+\frac{1-B}{B}+\frac{1-C}{C} \geq \frac{A}{2-A}+\frac{B}{2-B}+\frac{C}{2-C}.$

Or $\frac{1}{A}+\frac{1}{B}+\frac{1}{C} - 3 \geq \frac{A}{2-A}+\frac{B}{2-B}+\frac{C}{2-C}.$

I am stuck here, thought about using AM-GM-HM to get rid of the reciprocal on the RHS but it doesn't work if applied directly.

Answer 1

Hint Note $t\mapsto \frac1t$ is convex and $(1-a,1-b,1-c)\succ (\frac12+\frac a2, \frac12+\frac b2, \frac12+\frac c2)$, so it’s Karamata’s Inequality.

Answer 2

From Jensen's Inequality with the convex function $f(t):= \frac{1}{1-t}$ for $t\in (0,1)$, we have $$\frac{a}{1-c}f(a)+\frac{b}{1-c}f(b)\geq f\left(\frac{a^2}{1-c}+\frac{b^2}{1-c}\right)\,.$$ Since $f$ is increasing, the Power-Mean Inequality $\frac{a^2+b^2}{2}\geq \left(\frac{a+b}{2}\right)^2$ implies that $$f\left(\frac{a^2+b^2}{1-c}\right)\geq f\left(\frac{(a+b)^2}{2(1-c)}\right)=f\left(\frac{1-c}{2}\right)\,.$$ Thus, $$\frac{a}{1-a}+\frac{b}{1-b}\geq (1-c)\,f\left(\frac{a^2+b^2}{1-c}\right)\geq (1-c)\,f\left(\frac{1-c}{2}\right)=\frac{2(1-c)}{1+c}\,.$$ We have two more similar inequalities, and by summing all three of them (and dividing the result by $2$), we have $$\frac{a}{1-a}+\frac{b}{1-b}+\frac{c}{1-c}\geq \frac{1-a}{1+a}+\frac{1-b}{1+b}+\frac{1-c}{1+c}\,.$$ Adding $3$ to both sides of the inequality above, the required result is yielded. The equality case is, of course, $(a,b,c)=\left(\frac13,\frac13,\frac13\right)$.

Answer 3

Just another way, note for $x\in(0,1)$, $$f(x)=\frac1{1-x}-\frac2{1+x}-\frac98(3x-1)=\frac{(3x-1)^2(3x+1)}{8(1-x^2)}\geqslant0$$ while the inequality is just $f(a)+f(b)+f(c)\geqslant 0$.