Hy vọng a;b;c dương
Khi đó: \(\frac{a^2}{b^2}+1\ge\frac{2a}{b}\) ; \(\frac{b^2}{c^2}+1\ge\frac{2b}{c}\) ; \(\frac{c^2}{a^2}+1\ge\frac{2c}{a}\)
\(\Rightarrow\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}+3\ge2\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\)
\(\Leftrightarrow\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-3\right)\)
\(\Rightarrow\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+3\sqrt[3]{\frac{abc}{abc}}-3\)
\(\Rightarrow\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\)
Dấu "=" xảy ra khi \(a=b=c\)
