Do \(a,b,c>0\) nên theo quy tắc phân số: \(\dfrac{a}{a+b}< \dfrac{a+c}{a+b+c}\)
Tương tự: \(\dfrac{b}{b+c}< \dfrac{a+b}{a+b+c}\); \(\dfrac{c}{a+c}< \dfrac{b+c}{a+b+c}\)
\(\Rightarrow\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{a+c}< \dfrac{2\left(a+b+c\right)}{a+b+c}=2\)
Theo BĐT Cauchy: \(\sqrt{a\left(b+c\right)}\le\dfrac{a+b+c}{2}\Leftrightarrow\dfrac{2}{a+b+c}\le\dfrac{1}{\sqrt{a\left(b+c\right)}}\)
\(\Leftrightarrow\dfrac{2a}{a+b+c}\le\sqrt{\dfrac{a}{b+c}}\)
Tương tự \(\dfrac{2b}{a+b+c}\le\sqrt{\dfrac{b}{a+c}}\); \(\dfrac{2c}{a+b+c}\le\sqrt{\dfrac{c}{a+b}}\)
(3 dấu = không thể đồng thời xảy ra, để chặt chẽ bạn có thể chia trường hợp)
Cộng vế với vế:
\(\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{a+c}}+\sqrt{\dfrac{c}{a+b}}>\dfrac{2\left(a+b+c\right)}{a+b+c}=2\)
\(\Rightarrow\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{a+c}< \sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{a+c}}+\sqrt{\dfrac{c}{a+b}}\)