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Extension of the Triangle Inequality

  1. Oct 8, 2010 #1
    Pretty much knows the triangle inequality.
    [tex]\left| a + b \right| \le \left| a \right| + \left| b \right|[/tex]

    I was reading a source which asserted the following extension of the triangle inequality:
    [tex]\left| a + b \right|^p \le 2^p \left(\left| a \right|^p + \left| b \right|^p\right)[/tex]

    This is bugging me because I can't figure out how to prove or disprove it. It's sensible enough that it might actually be true... but want to know for sure whether it's true.

    I tried proving it by induction, but it got really messy...

    Is this inequality true? Or, is it wrong? Thanks!
     
  2. jcsd
  3. Oct 8, 2010 #2
    when you tried induction, did you try the binomial theorem?
     
  4. Oct 9, 2010 #3
    It follows from convexity of xp for p > 1.
    [tex]
    (\frac{1}{2}|a|+\frac{1}{2}|b|)^p\leq \frac{1}{2}|a|^p+\frac{1}{2}|b|^p
    [/tex]
    [tex]
    (|a|+|b|)^p\leq 2^{p-1}(|a|^p+|b|^p)
    [/tex]
     
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