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        general >> complex analysis >> true or false
        
(Message started by: tiox on Feb 6^th, 2006, 6:10pm)

Title: true or false
Post by tiox on Feb 6^th, 2006, 6:10pm

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?

Title: Re: true or false
Post by Eigenray on Feb 7^th, 2006, 9:26am

If f has no zeros on the unit disk, then it has a well-defined log f which is analytic there. In particular, [hide]its real part is harmonic[/hide].

In fact, if a₁, a₂, ..., 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, 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.