Question:
Let $a,b,c>0$,with $ab+bc+ac=1$ prove or disprove $$\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\ge\dfrac{4}{3}\cdot3^{\dfrac{1}{4}}\cdot a$$ or $$\left(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\right)^4\ge \dfrac{256}{27}a^4$$
I have used Cauchy-Schwarz inequality to prove $$\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\ge\dfrac{(a+b+c)^2}{2(ab+bc+ac)}=\dfrac{a^2+b^2+c^2}{2}+1\ge \dfrac{ab+bc+ac}{2}+1=\dfrac{3}{2}$$oh,That's is Nesbit inequality .I don't know what to do next.