$\sum_{cyc}\left(\frac{a^3+1}{a^2+1}\right) CAUCHY \geq \sum_{cyc}\left(\frac{a^2+1}{a+1} \right) CAUCHY \geq \sum_{cyc}{\frac{a+1}{2}}=\sum_{cyc}{a} \geq \frac{1}{27}(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})^4 $ Why does $a+b+c \geq \frac{1}{27}(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})^4$ because from am gm $a+b \geq 2\sqrt{ab}$ $\frac{1}{27}(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})^4 \leq (a+b+c)^4 * \frac{1}{(a+b+c)^3}=a+b+c $

It seems that a fourth power is missing, according to this or this.

I think It is the same just need a little modification $\sum_{cyc}\left(\frac{a^3+1}{a^2+1}\right)^4 CAUCHY \geq \sum_{cyc}\left(\frac{a^2+1}{a+1} \right)^4 CAUCHY \geq \sum_{cyc}{\frac{(a+1)^4}{16}} \geq \sum_{cyc}{a} \geq \frac{1}{27}(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})^4 $ why does $\sum_{cyc}{\frac{(a+1)^4}{16}} \geq \sum_{cyc}{a}$ because from cs-engel $\sum_{cyc}{\frac{(a+1)^4}{16}} \geq {\frac{((a+1)^2+(b+1)^2+(c+1)^2)^2}{48}} \geq a+b+c$ _____________________________________________________________________________________________________________________ Edit from my above post Why does $a+b+c \geq \frac{1}{27}(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})^4$ because from am gm $a+b \geq 2\sqrt{ab}$ $\frac{1}{27}(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})^4 \leq (a+b+c)^4 * \frac{1}{(a+b+c)^3}=a+b+c $