Xét \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge\frac{1}{\sqrt{ab}}+\frac{1}{\sqrt{bc}}+\frac{1}{\sqrt{ac}}=\sqrt{d}\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)\)
\(\Rightarrow ab+bc+ac\ge\frac{\sqrt{a}+\sqrt{b}+\sqrt{c}}{\sqrt{d}}\) và \(\frac{1}{1+ab+bc+ac}\le\frac{\sqrt{d}}{\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}}\)
Tương tự : \(\frac{1}{1+bc+cd+da}\le\frac{\sqrt{a}}{\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}}\)
\(\frac{1}{1+cd+da+ac}\le\frac{\sqrt{b}}{\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}}\)
\(\frac{1}{1+da+ab+bd}\le\frac{\sqrt{c}}{\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}}\)
Cộng theo vế ta được đpcm.