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Eigenray
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 Re: Maximum Modulus Theorem   « on: Apr 8th, 2009, 5:18pm » Quote Modify

Edit: original question (paraphrase) :

Suppose fn(z) is a sequence of analytic functions on the unit disk which converge uniformly on compact subsets to a non-zero function f.  If each fn has at most m zeroes (counting multiplicity), show that f has at most m zeroes.

Do you know Hurwitz's theorem?  It basically comes from the argument principle.
 « Last Edit: Apr 12th, 2009, 12:32pm by Eigenray » IP Logged
Eigenray
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 Re: Maximum Modulus Theorem   « Reply #1 on: Apr 8th, 2009, 9:09pm » Quote Modify

on Apr 8th, 2009, 8:00pm, trusure wrote:
 But I think it is not true if we say at least !!

Yes that's right; it's possible for some of the zeroes to wander off, as the example
fk(z) = z-1+1/k
shows.  But they can't suddenly appear in the limit.  If f had more than n zeros, we could fix a radius r < 1 within which f had more than n zeros, and eventually the fk would have the same number of zeros in that disk, a contradiction.
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Eigenray
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 Re: Maximum Modulus Theorem   « Reply #2 on: Apr 11th, 2009, 12:31am » Quote Modify

on Apr 10th, 2009, 9:29pm, trusure wrote:
 can we find a sequence {f_n} on B(0,1) that converges uniformly on compact subsets of B(0,1) to f(z) such that for all n f_n(z) has at least m zeros where as f(z) has exactly k zeros, for       0<= k<=m ??!

Sure, just take fn(z) = zk(z - (1-1/n))m-k.

Quote:
 i tried,, for k=0 we can take f_n(z)= (z^m)/n - 1/n^2 this has m roots in B(0,1), and f(z)=0 has zero roots, but in general ??

The zero function does not have zero roots!
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Grimbal
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 Re: Maximum Modulus Theorem   « Reply #3 on: Apr 12th, 2009, 7:53am » Quote Modify

Hello Eigenray,

Was there somebody else here, or are you having a talk with your imaginary friend?
 « Last Edit: Apr 12th, 2009, 9:26am by Grimbal » IP Logged
trusure
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 Re: Maximum Modulus Theorem   « Reply #4 on: Apr 12th, 2009, 8:09am » Quote Modify

ohh Sorry, I don't know how the questions deleted !? maybe by a mistake
I will post the problem again

sorry again
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trusure
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 Re: Maximum Modulus Theorem   « Reply #5 on: Apr 12th, 2009, 8:21am » Quote Modify

the problem was to find a sequnce of analytic functions as indicated above, and I was wondering that the sequence given by Mr. Eigenray has exactly m zeros inside B(0,1) not at least m zeros.

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Eigenray
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 Re: Maximum Modulus Theorem   « Reply #6 on: Apr 12th, 2009, 2:14pm » Quote Modify

If it has exactly m zeros then it also has at least m zeros.  I suppose you could give it m+1 zeros if you really wanted to.
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