I have been trying to minimum value of $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}$ for $a,b,c>0$ using Cauchy-schwarz and AM-GM-HM inequality, but I cannot seem to get a lower bound. I think the answer is $\frac{3}{2}$ when $a=b=c$, but using Cauchy-schwarz I got
$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq\frac{(\sqrt{a}+\sqrt{b}+\sqrt{c})^2}{2(a+b+c)}=\frac{a+b+c+2(\sqrt{ab}+\sqrt{bc}+\sqrt{ac})}{2(a+b+c)}$
I could not establish a bound for the last part of this inequality; it seems to be $\leq\frac{3}{2}$. Is there a lower bound?
