Proving $\frac{1}{16} \sum \frac{(b+c)(c+a)}{ab} +\frac{9}{4} \geq 4\sum \frac{ab}{(b+c)(c+a)}$

Question

For $a,b,c>0$. Prove: $$\frac{1}{16} \sum\limits_{cyc} {\frac { \left( b+c \right) \left( c+a \right) }{ba}}+\frac{9}{4} \geq 4\, \sum\limits_{cyc}{
\frac {ba}{ \left( b+c \right) \left( c+a \right) }}$$

My SOS's proof is:

It's equivalent to: $$\frac{1}{27}\sum\limits_{cyc} ab \left( a+b-8\,c \right) ^{2} \left( a+b-2\,c \right) ^{2}+\frac{26}{27}\sum\limits_{cyc}ab \left( a-b \right) ^{2} \left( a+b-2\,c \right) ^{2} +{\frac{50}{27}} \Big[\sum\limits_{cyc} a(b-c)^2\Big]^2 \geq 0$$

However, it's hard to find without computer.

So I'm looking for alternative solution without $uvw$. Thanks for a real lot!

Answer 1

We have$:$\begin{align*}\sum\limits_{cyc} {\frac { \left( b+c \right) \left( c+a \right) }{ba}}+36 - 64\, \sum\limits_{cyc}{ \frac {ba}{ \left( b+c \right) \left( c+a \right) }} =\frac{\prod \,(a+b)\sum\limits_{cyc} a(a-b)(a-c)+\Big[\sum\limits_{cyc} c(a-b)^2\Big]^2}{a\,b\,c\,(a+b)\,(b+c)\,(c+a)}\geq 0\end{align*} The last inequality is Schur degree $3,$ you can see many proof by SOS here.

Or this one is stronger$:$ here