Inequality: $(1+a)^7(1+b)^7(1+c)^7 > 7^7 a^4b^4c^4$

[kaliprasad]

if $a,b,c$ are positive real numbers show that $(1+a)^7(1+b)^7(1+c)^7 > 7^7 a^4b^4c^4$

 

[anemone]

Can you...hmm...

Spoiler

pretty please, give us some clues?

 

[kaliprasad]

Can you...hmm...

Spoiler

pretty please, give us some clues?

given up so early

Spoiler

multiply out the LHS

 

[anemone]

given up so early

Spoiler

You might be onto something!:p

But on the level, I have bestowed much thought on this problem...hehehe...

 

[Greg]

if $a,b,c$ are positive real numbers show that $(1+a)^7(1+b)^7(1+c)^7 > 7^7 a^4b^4c^4$

Spoiler

$$(1+a)(1+b)(1+c)=1+a+b+ab+c+ac+bc+abc$$

AM-GM inequality:

$$\dfrac{a+b+c+ab+ac+bc+abc}{7}\ge(abc)^{4/7}$$

$$\Rightarrow a+b+c+ab+ac+bc+abc\ge7(abc)^{4/7}$$

$$\Rightarrow1+a+b+ab+c+ac+bc+abc>7(abc)^{4/7}$$

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

$$\Rightarrow(1+a)^7(1+b)^7(1+c)^7>7^7a^4b^4c^4$$

$$\text{Q. E. D.}$$

 

[kaliprasad]

Spoiler

$$(1+a)(1+b)(1+c)=1+a+b+ab+c+ac+bc+abc$$

AM-GM inequality:

$$\dfrac{a+b+c+ab+ac+bc+abc}{7}\ge(abc)^{4/7}$$

$$\Rightarrow a+b+c+ab+ac+bc+abc\ge7(abc)^{4/7}$$

$$\Rightarrow1+a+b+ab+c+ac+bc+abc>7(abc)^{4/7}$$

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

$$\Rightarrow(1+a)^7(1+b)^7(1+c)^7>7^7a^4b^4c^4$$

$$\text{Q. E. D.}$$

good answer same as mine.

Spoiler

I hoped that Anemone would solve it particularly after my hint.