Results 1 to 2 of 2

Math Help - analytic functions, bounded domain

  1. #1
    Newbie
    Joined
    Dec 2008
    Posts
    17

    analytic functions, bounded domain

    Let f(z) and g(z) be analytic functions on the bounded domain D that extend continuously to \partial D and satisfy |f(z)+g(z)|<|f(z)|+|g(z)| on \partial D. Show that f(z) and g(z) have the same number of zeros in D, counting multiplicity.


    The hint here says that neither f(z) nor g(z) has zeros on \partial D, so each has at most finitely many zeros in D. Estimate shows \frac{f(z)}{g(z)} \not \in (-\infty, 0] for z near \partial D, so \text{Log}(f(z)/g(z)) is continuously defined near \partial D. Increase in \text{arg} f(z)/g(z) around any closed path near \partial D is zero.


    We have covered Rouche's Theorem too. I do not see how to prove this right now. I need some pointers on where to go. Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Quote Originally Posted by poincare4223 View Post
    Let f(z) and g(z) be analytic functions on the bounded domain D that extend continuously to \partial D and satisfy |f(z)+g(z)|<|f(z)|+|g(z)| on \partial D. Show that f(z) and g(z) have the same number of zeros in D, counting multiplicity.


    The hint here says that neither f(z) nor g(z) has zeros on \partial D, so each has at most finitely many zeros in D. Estimate shows \frac{f(z)}{g(z)} \not \in (-\infty, 0] for z near \partial D, so \text{Log}(f(z)/g(z)) is continuously defined near \partial D. Increase in \text{arg} f(z)/g(z) around any closed path near \partial D is zero.


    We have covered Rouche's Theorem too. I do not see how to prove this right now. I need some pointers on where to go. Thanks.
    Your hint gives that f and g have a finite number of 0's inside D, so pick a curve C that encloses them all (simple and closed) such that d(c,d) (where c\in C and d\in \partial D ) ensures that h(z)=\frac{f(z)}{g(z)} \notin [0,\infty ) (not the negative interval since what we have is |h(z)+1|<|h(z)|+1 ) and we define w(z)=\ln (h(z)) in this C (maybe what could cause problems in a rigorous proof is to build such a C because \partial D can be really complicated, but lets assume you can, or consider \partial D to be a simple closed curve) then w'(z)= \frac{f'(z)}{f(z)} -\frac{g'(z)}{g(z)} and then \int_{C} w'=0 but by the argument principle this integral is equal to Z_f-Z_g
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: March 5th 2012, 05:50 PM
  2. Analytic + bounded --> Uniformly continoius
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: February 6th 2011, 03:08 AM
  3. bounded domain, meromorphic functions
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: January 19th 2010, 05:40 PM
  4. Replies: 1
    Last Post: June 20th 2009, 09:27 PM
  5. Analytic in a Domain D
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 12th 2009, 06:46 PM

Search Tags


/mathhelpforum @mathhelpforum