\(\frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2}\ge\frac{a+b+c}{3}\)
\(\Leftrightarrow\frac{a^4}{a^3+a^2b+ab^2}+\frac{b^4}{b^3+b^2c+bc^2}+\frac{c^4}{c^3+c^2a+a^2c}\ge\frac{a+b+c}{3}\)
\(\Leftrightarrow\frac{\left(a^2\right)^2}{a^3+a^2b+ab^2}+\frac{\left(b^2\right)^2}{b^3+b^2c+bc^2}+\frac{\left(c^2\right)^2}{c^3+c^2a+a^2c}\ge\frac{a+b+c}{3}\)
Áp dụng bất đẳng thức cộng mẫu số cho vế trái
\(\Rightarrow\frac{\left(a^2\right)^2}{a^3+a^2b+ab^2}+\frac{\left(b^2\right)^2}{b^3+b^2c+bc^2}+\frac{\left(c^2\right)^2}{c^3+c^2a+a^2c}\ge\frac{\left(a^2+b^2+c^2\right)^2}{a^3+b^3+c^3+a^2b+ab^2+b^2c+bc^2+c^2a+a^2c}\)
\(\Rightarrow\frac{\left(a^2\right)^2}{a^3+a^2b+ab^2}+\frac{\left(b^2\right)^2}{b^3+b^2c+bc^2}+\frac{\left(c^2\right)^2}{c^3+c^2a+a^2c}\ge\frac{\left(a^2+b^2+c^2\right)^2}{\left(a^3+a^2b+a^2c\right)+\left(b^3+b^2c+ab^2\right)+\left(c^3+c^2a+bc^2\right)}\)
\(\Rightarrow\frac{\left(a^2\right)^2}{a^3+a^2b+ab^2}+\frac{\left(b^2\right)^2}{b^3+b^2c+bc^2}+\frac{\left(c^2\right)^2}{c^3+c^2a+a^2c}\ge\frac{\left(a^2+b^2+c^2\right)^2}{a^2\left(a+b+c\right)+b^2\left(a+b+c\right)+c^2\left(a+b+c\right)}\)
\(\Rightarrow\frac{\left(a^2\right)^2}{a^3+a^2b+ab^2}+\frac{\left(b^2\right)^2}{b^3+b^2c+bc^2}+\frac{\left(c^2\right)^2}{c^3+c^2a+a^2c}\ge\frac{\left(a^2+b^2+c^2\right)^2}{\left(a^2+b^2+c^2\right)\left(a+b+c\right)}=\frac{a^2+b^2+c^2}{a+b+c}\)
Chứng minh rằng: \(\frac{a^2+b^2+c^2}{a+b+c}\ge\frac{a+b+c}{3}\)
\(\Rightarrow3\left(a^2+b^2+c^2\right)\ge\left(a+b+c\right)^2\)
Áp dụng bất đẳng thức Bunhiacopski cho 3 bộ số thực không âm
\(\Rightarrow3\left(a^2+b^2+c^2\right)=\left(1+1+1\right)\left(a^2+b^2+c^2\right)\ge\left(a+b+c\right)^2\)( đpcm )
Vậy \(\frac{a^2+b^2+c^2}{a+b+c}\ge\frac{a+b+c}{3}\)
Vì \(\frac{\left(a^2\right)^2}{a^3+a^2b+ab^2}+\frac{\left(b^2\right)^2}{b^3+b^2c+bc^2}+\frac{\left(c^2\right)^2}{c^3+c^2a+a^2c}\ge\frac{a^2+b^2+c^2}{a+b+c}\)
\(\Rightarrow\frac{\left(a^2\right)^2}{a^3+a^2b+ab^2}+\frac{\left(b^2\right)^2}{b^3+b^2c+bc^2}+\frac{\left(c^2\right)^2}{c^3+c^2a+a^2c}\ge\frac{a+b+c}{3}\)
\(\Leftrightarrow\frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2}\ge\frac{a+b+c}{3}\) ( đpcm )