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Homework inequality -- Show that (a+1)(b+1)(c+1)(d+1) < 8(abcd+1)

  1. Mar 8, 2017 #1

    ssd

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    1. The problem statement, all variables and given/known data
    For a,b,c,d >1, Show that (a+1)(b+1)(c+1)(d+1) < 8(abcd+1)

    2. Relevant equations
    How to show this?

    3. 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.
     
    Last edited: Mar 8, 2017
  2. jcsd
  3. Mar 8, 2017 #2

    ssd

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    Can this be done by induction?
     
  4. Mar 8, 2017 #3
    (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 ?
     
  5. Mar 8, 2017 #4
    ssd, Buffu has given a nice lead. Recognize that ##ab>1## and ##cd>1##. So replace these in your expressions for ##a## and ##b##
     
  6. Mar 8, 2017 #5

    ssd

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    Thanks a lot to both of you.
     
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