Áp dụng bất đẳng thức Cauchy - Schwarz
\(\Rightarrow\left\{{}\begin{matrix}\dfrac{ab}{c}+\dfrac{bc}{a}\ge2\sqrt{\dfrac{ab^2c}{ca}}=2\sqrt{b^2}=2b\\\dfrac{bc}{a}+\dfrac{ca}{b}\ge2\sqrt{\dfrac{abc^2}{ab}}=2\sqrt{c^2}=2c\\\dfrac{ab}{c}+\dfrac{ca}{b}\ge2\sqrt{\dfrac{a^2bc}{bc}}=2\sqrt{a^2}=2a\end{matrix}\right.\)
\(\Rightarrow2\left(\dfrac{ab}{c}+\dfrac{bc}{a}+\dfrac{ca}{b}\right)\ge2\left(a+b+c\right)\)
\(\Rightarrow\dfrac{ab}{c}+\dfrac{bc}{a}+\dfrac{ca}{b}\ge a+b+c\) ( đpcm )
Dấu " = " xảy ra khi \(a=b=c\)
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