\(VT=\left(\dfrac{a}{1+4c^2}+\dfrac{b}{1+4a^2}+\dfrac{c}{1+4b^2}\right)+\left(\dfrac{1}{1+4c^2}+\dfrac{1}{1+4a^2}+\dfrac{1}{1+4b^2}\right)\)
\(VT=\dfrac{3}{2}-\left(\dfrac{4c^2a}{1+4c^2}+\dfrac{4a^2b}{1+4a^2}+\dfrac{4b^2c}{1+4b^2}\right)+3-\left(\dfrac{4c^2}{1+4c^2}+\dfrac{4a^2}{1+4a^2}+\dfrac{4b^2}{1+4b^2}\right)\)
Xét \(\dfrac{3}{2}-\left(\dfrac{4c^2a}{1+4c^2}+\dfrac{4a^2b}{1+4a^2}+\dfrac{4b^2c}{1+4b^2}\right)\)
Áp dụng bất đẳng thức Cauchy - Schwarz
\(\Rightarrow\left\{{}\begin{matrix}1+4c^2\ge2\sqrt{4c^2}=4c\\1+4a^2\ge2\sqrt{4a^2}=4a\\1+4b^2\ge2\sqrt{4b^2}=4b\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}\dfrac{4c^2a}{1+4c^2}\le\dfrac{4c^2a}{4c}=ca\\\dfrac{4a^2b}{1+4a^2}\le\dfrac{4a^2b}{4a}=ab\\\dfrac{4b^2c}{1+4b^2}\le\dfrac{4b^2c}{4b}=bc\end{matrix}\right.\)
\(\Rightarrow\dfrac{3}{2}-\left(\dfrac{4c^2a}{1+4c^2}+\dfrac{4a^2b}{1+4a^2}+\dfrac{4b^2c}{1+4b^2}\right)\ge\dfrac{3}{2}-\left(ab+bc+ca\right)\) (1)
Xét \(3-\left(\dfrac{4c^2}{1+4c^2}+\dfrac{4a^2}{1+4a^2}+\dfrac{4b^2}{1+4b^2}\right)\)
Áp dụng bất đẳng thức Cauchy - Schwarz
\(\Rightarrow\left\{{}\begin{matrix}1+4c^2\ge2\sqrt{4c^2}=4c\\1+4a^2\ge2\sqrt{4a^2}=4a\\1+4b^2\ge2\sqrt{4b^2}=4b\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}\dfrac{4c^2}{1+4c^2}\le\dfrac{4c^2}{4c}=c\\\dfrac{4a^2}{1+4a^2}\le\dfrac{4a^2}{4a}=a\\\dfrac{4b^2}{1+4b^2}\le\dfrac{4b^2}{4b}=b\end{matrix}\right.\)
\(\Rightarrow3-\left(\dfrac{4c^2}{1+4c^2}+\dfrac{4a^2}{1+4a^2}+\dfrac{4b^2}{1+4b^2}\right)\ge\dfrac{3}{2}\) (2)
Từ (1) và (2)
\(\Rightarrow VT\ge\dfrac{3}{2}-\left(ab+bc+ca\right)+\dfrac{3}{2}\)
\(\Rightarrow VT\ge3-\left(ab+bc+ca\right)\) (3)
Theo hệ quả của bất đẳng thức Cauchy
\(\Rightarrow\left(a+b+c\right)^2\ge3\left(ab+bc+ca\right)\)
\(\Rightarrow\dfrac{3}{4}\ge ab+bc+ca\)
\(\Rightarrow3-\dfrac{3}{4}\le3-\left(ab+bc+ca\right)\)
\(\Rightarrow\dfrac{9}{4}\le3-\left(ab+bc+ca\right)\) (4)
Từ (3) và (4)
\(\Rightarrow VT\ge\dfrac{9}{4}\)
\(\Leftrightarrow\dfrac{1+b}{1+4a^2}+\dfrac{1+c}{1+4b^2}+\dfrac{1+a}{1+4c^2}\ge\dfrac{9}{4}\) (đpcm)
Dấu " = " xảy ra khi \(a=b=c=\dfrac{1}{2}\)
