Đặt [tex]a=\frac{1}{x},b=\frac{1}{y},c=\frac{1}{z}[/tex] [tex]\Rightarrow abc\geq 1\Rightarrow \left\{\begin{matrix} ab\geq \frac{1}{c}\\ bc\geq \frac{1}{a}\\ ca\geq \frac{1}{b} \end{matrix}\right.[/tex]
Điều phải chứng minh [tex]\Leftrightarrow \frac{x}{y^3}+\frac{y}{z^3}+\frac{z}{x^3}\geq x+y+z[/tex] [tex]\Leftrightarrow \frac{b^3}{a}+\frac{c^3}{b}+\frac{a^3}{c}\geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}[/tex]
Ta có:[tex]\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}=\frac{a^4}{ab}+\frac{b^4}{bc}+\frac{c^4}{ca}\geq \frac{(a^2+b^2+c^2)^2}{ab+bc+ca}\geq \frac{(ab+bc+ca)^2}{ab+bc+ca}=ab+bc+ca\geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}[/tex](đpcm)