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Math Help - zeros of an analytic function

  1. #1
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    zeros of an analytic function

    let f, g be analytic on a domain containing a simple closed curve ɣ and its inside. Show that if |f(z)| > |g(z)| for all z in ɣ, then the two equations f(z)=g(z) and f(z)=0 have an equal number of solutions inside ɣ.

    I do not know how to prove this T-T plz help me!!!

    and use above to determine how many solutions are there for z^3 = z^2 + z + 6 in |z|<1. I know answer for this is "no solution" but why???

    plz!!!
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  2. #2
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    If |h-f|<|f|\;\forall z\in D then by Rouche, f and h have the same number of zeros in D counted with multiplicities. Now,if |g|<|f|, and we let h=f-g, then |h-f|<|f| and so by Rouche, h and f have the same number of zeros in D.

    In your case, you wish to know the number of zeros of z^3-z^2-z-6=0 in the unit disc. So what happens if we let h=z^3-z^2-z-6 and f=-6?
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  3. #3
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    thank so you much!!!!
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