Results 1 to 2 of 2

Math Help - unbounded component

  1. #1
    Junior Member
    Joined
    May 2008
    Posts
    50

    unbounded component

    If I have a function f non-vanishing, entire and non-constant, how can I demonstrate that each component of the set {z in \mathbb{C}; |f(z)|<1} is unbounded? If I assume that there is at least one bounded component of that set, how can I get a contradiction? I suppose that I should use Liouville's Theorem at somewhere...

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721

    Re: unbounded component

    If D is the unit disk, U a bounded component of f^{-1}(D) then f(U)=D (use the compactness of \overline{U}), now use that 0\notin f(U).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. component
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: February 18th 2011, 12:24 AM
  2. Unbounded?
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: February 7th 2010, 01:56 PM
  3. Unbounded Operators
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: March 22nd 2009, 02:01 PM
  4. Tangential component help
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 17th 2008, 12:46 PM
  5. Vector in Component
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: May 3rd 2008, 09:07 PM

Search Tags


/mathhelpforum @mathhelpforum