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Thread: bounded domain, meromorphic functions

  1. #1
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    bounded domain, meromorphic functions

    Let $\displaystyle D$ be a bounded domain, and let $\displaystyle f(z)$ and $\displaystyle h(z)$ be meromorphic functions on $\displaystyle D$ that extend to be analytic on $\displaystyle \partial D$. Suppose that $\displaystyle |h(z)|< |f(z)|$ on $\displaystyle \partial D$. Show by example that $\displaystyle f(z)$ and $\displaystyle f(z)+h(z)$ can have different numbers of zeros on $\displaystyle D$. What can be said about $\displaystyle f(z)$ and $\displaystyle f(z)+h(z)$? Prove your assertion.

    I do not see what example I can use here. I think what we can say is that they have the same number of zeros minus poles. I just don't see an example here. Thanks.
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  2. #2
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    Quote Originally Posted by hilbertcube182 View Post
    Let $\displaystyle D$ be a bounded domain, and let $\displaystyle f(z)$ and $\displaystyle h(z)$ be meromorphic functions on $\displaystyle D$ that extend to be analytic on $\displaystyle \partial D$. Suppose that $\displaystyle |h(z)|< |f(z)|$ on $\displaystyle \partial D$. Show by example that $\displaystyle f(z)$ and $\displaystyle f(z)+h(z)$ can have different numbers of zeros on $\displaystyle D$. What can be said about $\displaystyle f(z)$ and $\displaystyle f(z)+h(z)$? Prove your assertion.

    I do not see what example I can use here. I think what we can say is that they have the same number of zeros minus poles. I just don't see an example here. Thanks.
    How about $\displaystyle f(z)=\frac{2}{z^2}$ and $\displaystyle g(z)=\frac{1/2-z}{z^3}$ with $\displaystyle D=\{z\in \mathbb{C} : |z|< 1 \}$
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