Is it true that $\frac {a}{a+b}+\frac {b}{b+c}+\frac {c}{c+a}\geq \frac32$ if $a+b+c=1$?

Question

Is it possible to prove that $ \dfrac {a}{a+b}+\dfrac {b}{b+c}+\dfrac {c}{c+a}\geq \dfrac {3}{2}$, if $ a+b+c=1, a,b,c>0$?

Answer 1

I think it's wrong.

Try $c\rightarrow0^+$.

We obtain $$\frac{a}{a+b}\geq\frac{1}{2}$$ or $$a\geq b$$ and you can get a counterexample with $a<b$.

Also, we can see it for $a\rightarrow0^+$: we obtain $0\geq\frac{1}{2}.$

By the way, your second problem is true.

Indeed, we need to prove that $$\sum_{cyc}\frac{a^2+b}{b+c}\geq2$$ or $$\sum_{cyc}(a^2+b)(a+b)(a+c)\geq2\prod_{cyc}(a+b)$$ or $$\sum_{cyc}(a^2+b)(a^2+ab+ac+bc)\geq2\prod_{cyc}(a+b)$$ or $$\sum_{cyc}(a^2+b)(a+bc)\geq2\prod_{cyc}(a+b)$$ or $$\sum_{cyc}\left(a^3+\frac{1}{3}abc+ab+a^2b\right)\geq2\sum_{cyc}\left(a^2b+a^2c+\frac{2}{3}abc\right)$$ or $$\sum_{cyc}\left(a^3+\frac{1}{3}abc+a^2b+a^2c+abc+a^2b\right)\geq2\sum_{cyc}\left(a^2b+a^2c+\frac{2}{3}abc\right)$$ or $$\sum_{cyc}(a^3-a^2c)\geq0,$$ which is true by Rearrangement.

Answer 2

Michael is right, computing the left hand side minus the right hand side we obtain $$1/2\,{\frac { \left( b-c \right) \left( a-c \right) \left( a-b \right) }{ \left( a+b \right) \left( b+c \right) \left( c+a \right) }} \geq 0$$