# Inequalities Question

• Jun 20th 2009, 09:49 PM
Chandru1
Inequalities Question
If $\displaystyle a,b,c$ are positive real numbers prove that $\displaystyle \Bigl[ (1+a)(1+b)(1+c) \Bigr]^{7}>7^{7} a^{4}b^{4}c^{4}$
• Jun 20th 2009, 10:08 PM
simplependulum
Use AM $\displaystyle \geq$ GM . $\displaystyle a^7 + b^7 + c^7 + (ab)^7 + (ac)^7 + (bc)^7 + (abc)^7 \geq 7 (abc)^4$

Replace a,b,c by $\displaystyle a^{\frac{1}{7}}, b^{\frac{1}{7}}, c^{\frac{1}{7}}$
it becomes :

$\displaystyle a + b + c + ab + ac + bc + abc \geq 7 (abc)^{\frac{4}{7}}$

Therefore ,

$\displaystyle 1 + a + b + c + ab + ac + bc + abc \geq 1 + 7 (abc)^{\frac{4}{7}} > 7 (abc)^{\frac{4}{7}}$

Then ,

$\displaystyle (1+a)(1+b)(1+c) > 7 (abc)^{\frac{4}{7}}$

$\displaystyle [(1+a)(1+b)(1+c)]^7 > 7^7 (abc)^4$
• Jun 21st 2009, 01:09 AM
NonCommAlg
Quote:

Originally Posted by simplependulum
Use AM $\displaystyle \geq$ GM . $\displaystyle a^7 + b^7 + c^7 + (ab)^7 + (ac)^7 + (bc)^7 + (abc)^7 \geq 7 (abc)^4$

Replace a,b,c by $\displaystyle a^{\frac{1}{7}}, b^{\frac{1}{7}}, c^{\frac{1}{7}}$
it becomes :

$\displaystyle a + b + c + ab + ac + bc + abc \geq 7 (abc)^{\frac{4}{7}}$

Therefore ,

$\displaystyle 1 + a + b + c + ab + ac + bc + abc \geq 1 + 7 (abc)^{\frac{4}{7}} > 7 (abc)^{\frac{4}{7}}$

Then ,

$\displaystyle (1+a)(1+b)(1+c) > 7 (abc)^{\frac{4}{7}}$

$\displaystyle [(1+a)(1+b)(1+c)]^7 > 7^7 (abc)^4$

similarly, for any positive real numbers $\displaystyle x_1,x_2, \cdots , x_n$ we have: $\displaystyle (1+x_1)(1+x_2) \cdots (1+x_n) \geq 1 + (2^n - 1) \sqrt[2^n - 1]{(x_1x_2 \cdots x_n)^{2^{n-1}}}.$ as a result:

$\displaystyle [(1+x_1)(1+x_2) \cdots (1+x_n)]^{2^n - 1} > (2^n - 1)^{2^n - 1} (x_1x_2 \cdots x_n)^{2^{n-1}}.$