Question

cho a,b>0 chứng minh 1/(1+a)² + 1/(1+b)² ≥ 1/(1+ab)

Answer 1

Áp dụng BĐT Bunhiacopxki:

\(\left(1+ab\right)\left(1+\dfrac{a}{b}\right)\ge\left(1+a\right)^2\)

\(\Rightarrow\dfrac{1}{\left(1+a\right)^2}\ge\dfrac{1}{\left(1+ab\right)\left(1+\dfrac{a}{b}\right)}=\dfrac{b}{\left(a+b\right)\left(1+ab\right)}\)

Tương tự:

\(\dfrac{1}{\left(1+b\right)^2}\ge\dfrac{a}{\left(a+b\right)\left(1+ab\right)}\)

Cộng vế:

\(\dfrac{1}{\left(1+a\right)^2}+\dfrac{1}{\left(1+b\right)^2}\ge\dfrac{a+b}{\left(a+b\right)\left(1+ab\right)}=\dfrac{1}{1+ab}\)

Dấu "=" xảy ra khi \(a=b=1\)