I am trying to prove the following inequality for real numbers $a,b,c,d$ all of which are greater than $1$
$8(abcd+1) > (a+1)(b+1)(c+1)(d+1)$
I tried the following approaches :
Used the AM-GM inequality
Tried to form a polynomial whose roots are a,b,c,d
Tried the trigonometric substitution $a=\sec x_1$....
But still I couldn't get any closer to the answer.