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# Cauchy's Estimate

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1. I'm having trouble pinning down what should be a straightforward question.

Q: Let R>0 and f be an entire function such that for . Show that f is a polynomial of degree at most n.

I have a hint that says it would be helpful to apply Cauchy's estimate. So I fix R>0 and choose S>R. Then f is analytic on B(0,S) and for |z|>R. Then, applying Cauchy's estimate for any z0 in the annulus gives . Which implies f is poly of max degree n (as n'th derivative is constant).

But what about all those points inside B(0,R)? How do I deal with them?

2. Use Cauchy's integral formula to establish a bound on the function inside the disc, by integrating around a contour sufficiently large. (I think that's how it goes. If not I'll check my work and see how I went about it.)
3. (Original post by Zhen Lin)
Use Cauchy's integral formula to establish a bound on the function inside the disc, by integrating around a contour sufficiently large. (I think that's how it goes. If not I'll check my work and see how I went about it.)
I'm not entirely clear what you're getting at. Could we let, say, , and the conclude from Cauchy's integral formula (with for suitably large S>R) that ? Which gives us as an upper bound for the integral inside the disc. Is that enough to conclude that f itself is bounded inside the disc? Is that what you were suggesting?
4. Well, this is what I had in mind. Let , where and S is chosen so that for all t. Then , so , which you can now bound above by . So for , taking allows us to see that for as well. (Actually, this gives the result for all z, since a holomorphic function constant on some open subset of a connected domain is globally constant.)
5. Hmm, looks a bit heavy-handed but I can't see anything else so perhaps that's the best to be done. Thanks again, Zhen.

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Updated: May 27, 2010
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