BĐT $\Leftrightarrow (\frac{a+2}{4}-\frac{2a}{a+2})+(\frac{b+3}{4}-\frac{3b}{b+3})+(\frac{c+1}{4}-\frac{c}{c+1})\geq \frac{a+b+c+6}{4}-\frac{6}{a+b+c+6}$$\Leftrightarrow \frac{(a-2)^2}{a+2}+\frac{(b-3)^2}{b+3}+\frac{(c-1)^2}{c+1}\geq\geq \frac{(a+b+c-6)^2}{a+b+c+6} $
BĐT trên lđ theo $Cauchy-Schwart$
$\rightarrow $ đpcm!
Đẳng thức khi $\left\{ \begin{array}{l} a=2 \\b=3\\c=1 \end{array} \right../$