<s>[INST] Inequality with Four Numbers 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. 
 [/INST] $(a-1)(b-1)>0$ as $a,b>1 \implies ab+1>a+b $,  or $2(ab+1)>ab+1+a+b=(a+1)(b+1)$.  
Similarly $(c+1)(d+1)<2(cd+1)$. 
Now $ab,cd>1$ then by similar method $(ab+1)(cd+1)<2(abcd+1)$ hence $4(ab+1)(cd+1)<8(abcd+1) $ therefore $(a+1)(b+1)(c+1)(d+1)<8(abcd+1)$
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