Inegalitate in numere pozitive

Mateescu Constantin · Post by **Mateescu Constantin** » Sun Jun 07, 2009 4:34 pm

Fie \( a,\ b,\ c \) numere reale pozitive. Aratati ca

\( 4\le \frac{a+b+c}{\sqrt[3]{abc}}+\frac{8abc}{(a+b)(b+c)(c+a)}. \)

M.R. 3/2009

Marius Mainea · Post by **Marius Mainea** » Sun Jun 07, 2009 11:32 pm

\( RHS=\frac{a+b+c}{3\sqrt[3]{abc}}+\frac{a+b+c}{3\sqrt[3]{abc}}+\frac{a+b+c}{3\sqrt[3]{abc}}+\frac{8abc}{(a+b)(b+c)(c+a)}\ge4\sqrt[4]{\frac{(a+b+c)^38abc}{27abc(a+b)(b+c)(c+a)}}\ge LHS \) deoarece

\( (a+b)+(b+c)+(c+a)\ge 3\sqrt[3]{(a+b)(b+c)(c+a)} \)