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Topic: true or false (Read 3589 times) 

tiox
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true or false
« on: Feb 6^{th}, 2006, 6:10pm » 
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Could someone give me a hint here? Suppose a function f is analytic in a region which contains the closed unit disk, and satisfies f(0) = 1, and f>2 on the border of the disk. Must f have a zero inside the disk?


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Eigenray
wu::riddles Moderator Uberpuzzler
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Re: true or false
« Reply #1 on: Feb 7^{th}, 2006, 9:26am » 
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If f has no zeros on the unit disk, then it has a welldefined log f which is analytic there. In particular, its real part is harmonic. In fact, if a_{1}, a_{2}, ..., a_{n} are its zeros in the disk, then one can show [sum] log(1/a_{i}) > log 2. Hence there's at least one zero, and in fact there must be at least one within the disc of radius 2^{1/n} < 1. This is but a special case of Jensen's (or more generally, PoissonJensen's) formula, which shows how the growth of a function is directly tied to the distribution of its zeros and poles, making it a key ingredient in the study of entire functions and Nevanlinna theory, for example.


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