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Proving an inequality

  1. Aug 14, 2011 #1
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Aug 14, 2011 #2

    SammyS

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    Show what you have tried, and where you got stuck.

    That way it will be easier for use to give the appropriate help.
     
  4. Aug 14, 2011 #3
    Don't forget about the reverse triangle inequality: |x - y| >= ||x| - |y||
     
  5. Aug 15, 2011 #4
    Last edited by a moderator: May 5, 2017
  6. Aug 15, 2011 #5
    It will be easier to break this into two problems. First prove that [itex]\left| sin z \right| \leq \frac{e^{y} + e^{-y}}{2}[/itex] , then prove that [itex]\left| sin z \right| \geq \frac{e^{\left| y \right|} + e^{- \left| y \right|}}{2}[/itex].

    Also, use [itex] \left| sin z \right| = \frac{e^{i(x+iy)} - e^{-i(x+iy)}}{2i} [/itex] .
     
  7. Aug 16, 2011 #6
    That's what i did. can you please look at my answer? isn't it correct?
    thank you.
     
  8. Aug 16, 2011 #7
    Sorry, I see that you did write [itex] \left| sin z \right| = \frac{e^{i(x+iy)} - e^{-i(x+iy)}}{2i} [/itex]. However, I don't understand how you came up with your answer: where did the absolute value signs in [itex] \left| \frac{e^{y} + e^{-y}}{2} \right| [/itex] emerge from? You can't just insert them. And how did you re-arrange the inequality? Was there a typo in your original statement of the problem? In any case, your answer doesn't prove the inequality, because I cannot see its validity just by looking at it. With these types of problems, you really need to break it down to something like [itex] -e^{-y} \leq e^{-y} \; \Rightarrow \; -1 \leq 1 [/itex], which we can all agree is true. Also, in proofs you *need* to show your steps, and you always should here anyways so we can help you out.

    Alright, let's try doing this one part at a time. First, prove that:

    [itex] \left| sin z \right| \leq \frac{e^{y} + e^{-y}}{2} [/itex], or

    [itex] \left| \frac{e^{i(x+iy)} - e^{-i(x+iy)}}{2i} \right| \leq \frac{e^{y} + e^{-y}}{2} [/itex]. Cancel the 2's, multiply by i/i and rearrange exponentials on the left,

    [itex] \left| -i e^{ix} e^{-y} + i e^{-ix} e^y \right| \leq e^{y} + e^{-y} [/itex] ....Now try applying the triangle inequality to this. After proving this, a similar approach is used for [itex] \left| sin z \right| \geq \frac{e^{\left| y \right|} + e^{- \left| y \right|}}{2} [/itex]
     
    Last edited: Aug 16, 2011
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