If a,b,c are positive real numbers, then prove that ${[(1 + a) (1 + b) (1 + c)]}^{7} > 7^{7} a^{4} b^{4} c^{4}$ .

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Solution

${[(1 + a) (1 + b) (1 + c)]}^{7} > 7^{7} a^{4} b^{4} c^{4}$
i.e., $(1 + \sum a + \sum a b + a b c) > 7 {(a b c)}^{4 / 7}$
Since, $A M > G M$
$a + b + c + \geq 3 {(a b c)}^{1 / 3}$
$a b + b c + a c \geq 3 {(a^{2} b^{2} c^{2})}^{1 / 3}$
Hence,
$1 + \sum a + \sum a b + a b c \geq 1 + 3 {(a b c)}^{1 / 3} + 3 {(a b c)}^{2 / 3} + a b c$ $1 + \sum a + \sum a b + a b c \geq 1 + {(a b c)}^{1 / 3} + {(a b c)}^{1 / 3} + {(a b c)}^{1 / 3} + {(a b c)}^{2 / 3} + {(a b c)}^{2 / 3} + {(a b c)}^{2 / 3} + a b c$
So,
LHS $\geq 7 {[1 {(a b c)}^{1 / 3} {(a b c)}^{1 / 3} {(a b c)}^{1 / 3} {(a b c)}^{2 / 3} {(a b c)}^{2 / 3} {(a b c)}^{2 / 3} (a b c)]}^{1 / 7}$ $\geq 7 {[{(a b c)}^{9 / 3} (a b c)]}^{1 / 7}$
$\geq 7 {(a b c)}^{4 / 7}$

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