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Complex Analysis: Maximal Modulus Principle question

Hi all,

I am stuck with this problem

Attachment 22252

It is suggested that the Maximal Modulus Principle would help but in fact, I couldn't find a place to apply it! Anyone can suggest me how to approach this problem because I am really lost now. Thanks a lot.

Re: Complex Analysis: Maximal Modulus Principle question

I've been thinking of this problem but I just can't complete a proof, here's a possible route (hopefully) towards a solution: By a simple continuity argument there is an such that on we have that does tend to zero. Now assume the following is true

**Claim:** If on the line with then on

then by an argument identical to the first on so the set on which has to be . I'm having a little trouble wih the claim though (particularly estimating in the verticla boundary of the set for some ; this is enough by the MMP).

Sorry I can't be of more help, if you find a proof please post it here.

Re: Complex Analysis: Maximal Modulus Principle question

There's probably a nicer way than this (i.e. a way that invokes maximum modulus at least). But for now I don't see why this doesn't work.

Take the sequence of points . This sequence is in *D*.

By continuity of *f* we can flex the limit in and out of the function:

Now *f* is holomorphic and bounded on *D*, so it has no singularity at infinity (in particular, no erratic essential singularity behavior). So

exists. So all unbounded sequences in *D* tend to this limit. We just showed one such sequence tends to *A*.

(So *A *is what all unbounded sequences in *D* tend to*. *In particular, any unbounded sequence with fixed imaginary component between 0 and 1 tends to *A.*)

Re: Complex Analysis: Maximal Modulus Principle question

Quote:

Originally Posted by

**gosuman** Now

*f* is holomorphic and bounded on

*D*, so it has no singularity at infinity (in particular, no erratic essential singularity behavior). So

exists.

This I don't get, isn't this even stronger than the claim to be proved: We know that the function tends to a limit along the reals so by these two lines the problem becomes trivial.

More to the point, can you prove your statement above?