Giả sử b nằm giữa a và c, dễ cm: [tex]a^2b+b^2c+c^2a\leq b(a^2+ac+c^2)[/tex]
[tex]\rightarrow a^2b+b^2c+c^2a-abc\leq b(a^2+ac+c^2)-abc=b(a^2+c^2)[/tex]
[tex]=2\sqrt{b^2.\frac{a^2+c^2}{2}.\frac{a^2+c^2}{2}}\leq 2\sqrt{\left ( \frac{b^2+2.\frac{a^2+c^2}{2}}{3} \right )^3}=2[/tex]
[tex]\rightarrow a^2b+b^2c+c^2a \leq 2+abc \leq 2+\frac{(a+b+c)^3)}{27}=3[/tex]