Áp dụng Holder:
\(24VT=\left(1+1+1+1+1+1\right)\left(a^3+a^3+c^3+c^3+b^3+b^3\right)\left(\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{a^3}+\frac{1}{c^3}\right)\ge\left(\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\right)^3\)
Mà \(\left(\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\right)^2\ge36\)( AM-GM)
\(24VT\ge36\left(\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\right)\Leftrightarrow VT\ge VF\)
Dấu = xảy ra khi a=b=c .
P/s: BĐT holder: \(\left(a_1^n+a^n_2+...a_3^n\right)\left(b_1^n+b_2^n+...b_n^n\right)...\left(z_1^n+z_2^n+...z_n^n\right)\ge\left(a_1.b_1..z_1+a_2.b_2..z_2+...+a_n.b_n.z_n\right)^n\)