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tiox
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« on: Feb 6th, 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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« Reply #1 on: Feb 7th, 2006, 9:26am »
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If f has no zeros on the unit disk, then it has a well-defined log f which is analytic there.  In particular, its real part is harmonic.
 
In fact, if a1, a2, ..., an are its zeros in the disk, then one can show
[sum] log(1/|ai|) > 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, Poisson-Jensen'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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