Re: Catheodory's inequality help please

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 10/16/04 

Date: Fri, 15 Oct 2004 19:47:58 -0500

On Fri, 15 Oct 2004 07:43:23 -0400, "Isaac" <Isharu@yahoo.com> wrote:

>
>"David C. Ullrich" <ullrich@math.okstate.edu> wrote in message
>news:217vm0pgor0si3ncamujcr3q5fj3n3d7uq@4ax.com...
>> On Thu, 14 Oct 2004 17:37:33 -0400, "Isaac" <Isharu@yahoo.com> wrote:
>>
>>>How can one prove the following Catheodory inequality :
>>>
>>>If f is analytic on the closed disk cl(B(0;R)) and M(r) = max {|f(z)| :
>>>|z|
>>>= r}, A(r) = max {Re f(z) : |z| = r}, then for 0 < r < R, if A(r) >= 0,
>>>
>>>M(r) <= ( [R+r] / [R-r] ) * [A(R) + |f(0)|].
>>>
>>>There is a hint that says consider the case where f(0) = 0 and examine the
>>>function g(z) = f(Rz) [2A(R) + f(Rz)]^{-1} for |z| < 1.
>>>
>>>Well I examined that, and found that |g(z)| <= 1. Also, g(0) = 0, and so
>>>therefore we can apply the Schwartz lemma so |g(z)| <= |z|.
>>>
>>>But what now?[...]
>>
>> Well you're way ahead of me on this one - I don't see why |g(z)| <= 1.
>> (Did you state the problem and the hint correctly?)
>>
>
>Yes, stated correctly. Actually, I am wrong about |g(z)| <= 1 (well at
>least my reasoning was wrong). So I don't
>see |g(z)| <= 1 anymore. The hint was given as it is stated(I'll note it
>says "First consider the case ...")
>so I assume author wants me to deduce something from first considering this
>case. I guess the problem
>is that I can't get anything out of this case right now. It looks like a
>combination of a Schwarz lemma and
>Maximum modulus principles.
>
>As well, I thought g(0) = 0, but now I don't see that anymore. So I am back
>to square one. Any ideas?

Well I don't see this right now, sorry. If A(r) were the maximum of
the absolute value of the real part I could imagine proving an
inequality like this by methods having nothing to do with the
hint - if g had -2A + f in the denominator instead of what you
say I could imagine maybe something... sorry.

************************

David C. Ullrich