Question

Cho a,b,c\(\ge1\)CMR \(\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\ge\frac{1}{1+\sqrt[4]{ab^3}}+\frac{1}{1+\sqrt[4]{bc^3}}+\frac{1}{1+\sqrt[4]{ca^3}}\)

Answer 1

Ta có \(a+b+b+b\ge4\sqrt[4]{abbb}\)(theo BĐT Cosi)

\(\Leftrightarrow a+3b\ge\sqrt[4]{ab^3}\)

\(\Leftrightarrow\frac{a+3b}{4}\ge4\sqrt[4]{ab^3}\)

Mà \(a,b,c\ge1\Rightarrow a+3b\ge4\Rightarrow\frac{a+3b}{4}\ge1\)

\(\Leftrightarrow1+\sqrt[4]{ab^3}\ge1+a\)

\(\Rightarrow\frac{1}{1+\sqrt[4]{ab^3}}\le\frac{1}{1+a}\left(1\right)\)

Tương tự \(\hept{\begin{cases}\frac{1}{1+\sqrt[4]{bc^3}}=\frac{1}{1+b}\left(2\right)\\\frac{1}{1+\sqrt[4]{ca^3}}=\frac{1}{1+c}\left(3\right)\end{cases}}\)

(1) (2) (3) => \(\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\ge\frac{1}{1+\sqrt[4]{ab^3+1}}+\frac{1}{1+\sqrt[4]{bc^3}}+\frac{1}{1+\sqrt[4]{ca^3}}\)(đpcm)