Results 1 to 4 of 4

Math Help - Maximum Modulus Theorem

  1. #1
    Member
    Joined
    Oct 2009
    From
    United States
    Posts
    169

    Maximum Modulus Theorem

    Suppose that f is entire and that |f(z)-1|<1 for all z with |z|=1. Prove that f(z) has no zeroes in the disk \Delta= \{z:|z|<1\} .
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45

    Re: Maximum Modulus Theorem

    Quote Originally Posted by tarheelborn View Post
    Suppose that f is entire and that |f(z)-1|<1 for all z with |z|=1. Prove that f(z) has no zeroes in the disk \Delta= \{z:|z|<1\} .
    Hint Denote g(z)=f(z)-1 and apply the Rouche's Theorem to f(z)=g(z)+1 .
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2009
    From
    United States
    Posts
    169

    Re: Maximum Modulus Theorem

    We haven't had Rouche's Theorem. I have the Maximum Modulus Theorem and I am not really sure how to apply that here.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Mar 2010
    From
    Beijing, China
    Posts
    293
    Thanks
    23

    Re: Maximum Modulus Theorem

    The maximum modlus theorem states if g is holomorphic, |g| reaches its maximum only on the boundary.
    Now g=f-1 is holomorphic, and |g|<1 on |z|=1, so |g|<1 in the disk |z|<1.
    So |f|=|1+g|>=|1-|g||>|1-1|=0 in the disk.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: December 5th 2011, 10:47 PM
  2. Replies: 3
    Last Post: December 4th 2011, 03:30 PM
  3. Maximum modulus and mean values
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: November 24th 2009, 09:49 AM
  4. by using Maximum AND minimum modulus principle
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: August 5th 2009, 12:22 AM
  5. Application of Maximum Modulus Principle on the Boundary
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 21st 2009, 12:31 PM

Search Tags


/mathhelpforum @mathhelpforum