\(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{9}{a+b+c}\ge4\left(\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{a+c}\right)\)
\(\Leftrightarrow\dfrac{a+b+c}{a}+\dfrac{a+b+c}{b}+\dfrac{a+b+c}{c}+9\) \(\ge4\left(a+b+c\right)\left(\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{a+c}\right)\)
\(\Leftrightarrow\dfrac{b+c}{a}+\dfrac{a+c}{b}+\dfrac{a+b}{c}+12\ge4\left(3+\dfrac{c}{a+b}+\dfrac{a}{b+c}+\dfrac{b}{a+c}\right)\)
\(\Leftrightarrow\dfrac{b+c}{a}+\dfrac{a+c}{b}+\dfrac{a+b}{c}\ge4\left(\dfrac{c}{a+b}+\dfrac{a}{b+c}+\dfrac{b}{a+c}\right)\).
Áp dụng bất đẳng thức \(\dfrac{1}{a+b}\le\dfrac{1}{4}\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\) ta có:
\(\dfrac{c}{a+b}+\dfrac{a}{b+c}+\dfrac{b}{a+c}\le\dfrac{1}{4}\left(\dfrac{c}{a}+\dfrac{c}{b}+\dfrac{a}{b}+\dfrac{a}{c}+\dfrac{b}{a}+\dfrac{b}{c}\right)\) \(=\dfrac{1}{4}\left(\dfrac{b+c}{a}+\dfrac{a+c}{b}+\dfrac{a+b}{c}\right)\).
Suy ra \(4\left(\dfrac{c}{a+b}+\dfrac{a}{b+c}+\dfrac{b}{a+c}\right)\le\dfrac{b+c}{a}+\dfrac{a+c}{b}+\dfrac{a+b}{c}\) 9 (đpcm).
