Show (a+1)(b+1)(c+1)(d+1) < 8(abcd+1) for a,b,c,d > 1

[ssd]

Homework Statement

For a,b,c,d >1, Show that (a+1)(b+1)(c+1)(d+1) < 8(abcd+1)

Homework Equations

How to show this?

The Attempt at a Solution

I could show for two variables, (a+1)(b+1)<2(ab+1). Tried C-S, AM-GM inequalities in different form and variable transformations. But still no result. It's my daughters test question. Any help is appreciated.

 

[ssd]

Can this be done by induction?

 

[Buffu]

(a+1)(b+1)<2(ab+1)
(d+1)(c+1)<2(dc+1)

(a+1)(b+1)(d+1)(c+1)< 4(ab+1)(dc+1)

Can you complete now ?

 

[issacnewton]

ssd, Buffu has given a nice lead. Recognize that $ab>1$ and $cd>1$. So replace these in your expressions for $a$ and $b$

 

[ssd]

Thanks a lot to both of you.