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Thread: analytic functions, bounded domain

  1. #1
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    analytic functions, bounded domain

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


    The hint here says that neither $\displaystyle f(z)$ nor $\displaystyle g(z)$ has zeros on $\displaystyle \partial D$, so each has at most finitely many zeros in $\displaystyle D$. Estimate shows $\displaystyle \frac{f(z)}{g(z)} \not \in (-\infty, 0]$ for z near $\displaystyle \partial D$, so $\displaystyle \text{Log}(f(z)/g(z))$ is continuously defined near $\displaystyle \partial D$. Increase in $\displaystyle \text{arg} f(z)/g(z)$ around any closed path near $\displaystyle \partial D$ 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.
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  2. #2
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    Quote Originally Posted by poincare4223 View Post
    Let $\displaystyle f(z)$ and $\displaystyle g(z)$ be analytic functions on the bounded domain $\displaystyle D$ that extend continuously to $\displaystyle \partial D$ and satisfy $\displaystyle |f(z)+g(z)|<|f(z)|+|g(z)|$ on $\displaystyle \partial D$. Show that $\displaystyle f(z)$ and $\displaystyle g(z)$ have the same number of zeros in D, counting multiplicity.


    The hint here says that neither $\displaystyle f(z)$ nor $\displaystyle g(z)$ has zeros on $\displaystyle \partial D$, so each has at most finitely many zeros in $\displaystyle D$. Estimate shows $\displaystyle \frac{f(z)}{g(z)} \not \in (-\infty, 0]$ for z near $\displaystyle \partial D$, so $\displaystyle \text{Log}(f(z)/g(z))$ is continuously defined near $\displaystyle \partial D$. Increase in $\displaystyle \text{arg} f(z)/g(z)$ around any closed path near $\displaystyle \partial D$ 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 $\displaystyle 0$'s inside $\displaystyle D$, so pick a curve $\displaystyle C$ that encloses them all (simple and closed) such that $\displaystyle d(c,d)$ (where $\displaystyle c\in C$ and $\displaystyle d\in \partial D$ ) ensures that $\displaystyle h(z)=\frac{f(z)}{g(z)} \notin [0,\infty )$ (not the negative interval since what we have is $\displaystyle |h(z)+1|<|h(z)|+1$ ) and we define $\displaystyle w(z)=\ln (h(z))$ in this $\displaystyle C$ (maybe what could cause problems in a rigorous proof is to build such a $\displaystyle C$ because $\displaystyle \partial D$ can be really complicated, but lets assume you can, or consider $\displaystyle \partial D$ to be a simple closed curve) then $\displaystyle w'(z)= \frac{f'(z)}{f(z)} -\frac{g'(z)}{g(z)}$ and then $\displaystyle \int_{C} w'=0$ but by the argument principle this integral is equal to $\displaystyle Z_f-Z_g$
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