Complex Analysis and Analytic Functions

[bballife1508]

Let f be analytic for |z| less than or equal to 1 and suppose that |f(z)| less than or equal to |e^z| when |z|=1. Show

(a)|f(z)| less than or equal to |e^z| when |z|<1

and

(b)If f(0)=i, then f(z)=ie^z for all z with |z| less than or equal to 1

 

[Office_Shredder]

What do you think about the problem?

 

[bballife1508]

I honestly have no idea where to even start.

 

[bballife1508]

Well i don't even know what combinatorics is so can you help me understand the algebraic way?

 

[snipez90]

This has nothing to do with combinatorics. What theorems have you learned in complex analysis that deal with inequalities that involve values of a holomorphic function on the boundary?

 

[bballife1508]

holomorphic function?

 

[snipez90]

Yeah don't worry about that, the words holomorphic and analytic are interchangeable (roughly, complex differentiability of f is equivalent to f having a local power series representation). The point is you need to look through your textbook or notes for one or two particular theorems that deal with this exact situation described in the problem.

 

[bballife1508]

for part (a) I am given a hint to consider f(z)/e^z... Do you know what I can do with this?

 

[snipez90]

Well, it might simplify things a little. The idea is to consider the modulus of that, but this alone doesn't get you very far (unless there is an easy way to work from first principles that I'm overlooking...). So you already know Liouville, what other big theorems have you learned in complex?

 

[bballife1508]

I'm pretty sure we covered just about all there is, cauchy liouville, morera, green

 

[snipez90]

Hmm okay, well my hint for that one problem where I asked you to look through your post history was Cauchy's inequalities, and you might be able to apply that here, barring any oversights on my part (I've never used it before in this capacity, but at the moment I don't see why not). Have you heard of the maximum modulus principle or Schwarz's lemma before?

 

[bballife1508]

yes we covered that in class, the maximum modulus

 

[snipez90]

Okay, I'm going to leave you to think about this for a bit, as I've suggested a few approaches. The maximum modulus principle implies something specific about where the maximum of an analytic function is obtained (look this up if you need to). Try to use this in combination with the hint.

 

[bballife1508]

the maximum must be on the boundary correct?

 

[snipez90]

Yes, now define $g(z) = f(z)/e^z$ as suggested in the hint. The hypothesis tells us $|f(z)| \leq |e^z|$ when |z| = R. So what is |g(z)| less than or equal to on |z| = R? You really need to think through this, and figure out why part a) is related to the fact that the maximum of an analytic function is on the boundary. Explain as much as you have figured out before asking a question, because you will get points on the final for doing that.

 

[bballife1508]

|g(z)| should be less than or equal to 1 since |f(z)| is less than or equal to |e^z|

 

[snipez90]

Yes, now part a) asks why $|g(z)| \leq 1$ on |z| < 1, given that we already know it holds on |z| = 1. Geometrically, what region is |z| < 1? |z| = 1? What is the boundary here?

 

[bballife1508]

g(z) is also analytic so it too achieves its maximum on the boundary which would mean that g(z) is less than or equal to 1 inside as well which proves that |f(z)| is less than or equal to |e^z| inside... I'm not sure if this is correct, but i feel like I'm on the right track

 

[bballife1508]

the boundary is the unit circle

 

[bballife1508]

a little help with part b) would be great. i understand why it is true but do not know exactly what to say about it.