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

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The inequality $(1+a)^7(1+b)^7(1+c)^7 > 7^7 a^4b^4c^4$ is to be proven for positive real numbers $a$, $b$, and $c$. Participants in the discussion agree on the validity of the inequality and some provide similar reasoning in their proofs. The conversation reflects a collaborative effort to understand and validate the mathematical expression. The focus remains on demonstrating the inequality through various approaches. The discussion highlights the importance of mathematical inequalities in understanding relationships between variables.
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if $a,b,c$ are positive real numbers show that $(1+a)^7(1+b)^7(1+c)^7 > 7^7 a^4b^4c^4$
 
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Can you...hmm...

pretty please, give us some clues?:o
 
anemone said:
Can you...hmm...

pretty please, give us some clues?:o

given up so early

multiply out the LHS
 
kaliprasad said:
given up so early

You might be onto something!:p

But on the level, I have bestowed much thought on this problem...hehehe...
 
kaliprasad said:
if $a,b,c$ are positive real numbers show that $(1+a)^7(1+b)^7(1+c)^7 > 7^7 a^4b^4c^4$

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

AM-GM inequality:

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

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

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

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

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

$$\text{Q. E. D.}$$
 
greg1313 said:
$$(1+a)(1+b)(1+c)=1+a+b+ab+c+ac+bc+abc$$

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