<s>[INST] Maximum and minimum value of an inequality Hello during a problem I have to solve this with $a,b,c$ positiv real number:
$$6\geq\frac{a^2 + a b + 2 a + b^2 + 3}{a^2 + a b - 2 a + b^2 + 3}+\frac{c^2 + c b + 2 b + b^2 + 3}{c^2 + c b - 2 b + b^2 + 3}+\frac{a^2 + a c + 2 c + c^2 + 3}{a^2 + a c - 2 c + c^2 + 3}\geq 3$$
Edit:Wolfram alpha says that the minimum value is 3.
Second edit :This is what I want to solve :

$$\sum_{cyc}\frac{a^2 + a b + 2 a + b^2 + 3}{a^2 + a b - 2 a + b^2 + 3}\leq \sum_{cyc}\frac{a + \sqrt{a b} + 2 \sqrt{a} + b + 3}{a + \sqrt{a b}- 2 \sqrt{a} + b + 3}$$

I try many classical inequalities , but without success .
Thanks a lot .
 [/INST] The inequality

$$\sum_{cyc}\frac{a^2 + a b + 2 a + b^2 + 3}{a^2 + a b - 2 a + b^2 + 3}\leq \sum_{cyc}\frac{a + \sqrt{a b} + 2 \sqrt{a} + b + 3}{a + \sqrt{a b}- 2 \sqrt{a} + b + 3}.$$

is wrong. 
Try $b=c\rightarrow0^+$ and $a=1.1$.
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