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Math Help - bounded domain, meromorphic functions

  1. #1
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    bounded domain, meromorphic functions

    Let D be a bounded domain, and let f(z) and h(z) be meromorphic functions on D that extend to be analytic on \partial D. Suppose that |h(z)|< |f(z)| on \partial D. Show by example that f(z) and f(z)+h(z) can have different numbers of zeros on D. What can be said about f(z) and 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 D be a bounded domain, and let f(z) and h(z) be meromorphic functions on D that extend to be analytic on \partial D. Suppose that |h(z)|< |f(z)| on \partial D. Show by example that f(z) and f(z)+h(z) can have different numbers of zeros on D. What can be said about f(z) and 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 f(z)=\frac{2}{z^2} and g(z)=\frac{1/2-z}{z^3} with D=\{z\in \mathbb{C} : |z|< 1 \}
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