Thread: Complex Analysis: Maximal Modulus Principle question

1. Complex Analysis: Maximal Modulus Principle question

Hi all,

I am stuck with this problem

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.

2. 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 $r>0$ such that on $0\leq Im(z) \leq r$ we have that $f$ does tend to zero. Now assume the following is true

Claim: If $f\to 0$ on the line $Im(z)=s$ with $0 then $f\to 0$ on $0\leq Im(z) \leq s$

then by an argument identical to the first $f\to 0$ on $0\leq Im(z) \leq s+r_1$ so the set on which $f\to 0$ has to be $D$. I'm having a little trouble wih the claim though (particularly estimating $f$ in the verticla boundary of the set $0\leq Im(z) \leq s, Re(z)>R$ for some $R>0$; this is enough by the MMP).

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

3. 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 $\left\{a_n = n+i\left(\frac{1}{n}\right)\right\}_{n=2}^\infty$. This sequence is in D.

By continuity of f we can flex the limit in and out of the function:
$\lim_{n \to \infty} f(a_n) = f\left(\lim_{n \to \infty} a_n\right) = f\left(\lim_{n \to \infty} n + \lim_{n \to \infty} i\left(\frac{1}{n}\right)\right) = f\left(\lim_{n \to \infty} n\right)= f\left(\lim_{x \to \infty} x\right) = \lim_{x \to \infty} f(x) = A.$

Now f is holomorphic and bounded on D, so it has no singularity at infinity (in particular, no erratic essential singularity behavior). So
$\lim_{z \to \infty}f(z)$ 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.)

4. Re: Complex Analysis: Maximal Modulus Principle question

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
$\lim_{z \to \infty}f(z)$ 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?