Let be analytic in the disk . If has a zero of order 2 at the origin and in that disk. Prove that in

I have no idea where to start. Thank you.

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- Dec 16th 2011, 04:13 AMfareastmovementAnalytic problem
Let be analytic in the disk . If has a zero of order 2 at the origin and in that disk. Prove that in

I have no idea where to start. Thank you. - Dec 16th 2011, 04:35 AMFernandoRevillaRe: Analytic problem
: In the series expansion we have , so is analytic in .__Hint__ - Dec 16th 2011, 06:07 AMfareastmovementRe: Analytic problem
Thank you for you reply, I could do what you said but whats the next step? Thanks:)

- Dec 16th 2011, 07:31 AMFernandoRevillaRe: Analytic problem
By hypothesis so, for . This equality is also valid for according to the Maximum Modulus Principle. If we fix in we have for all and . You can conclude.

- Dec 16th 2011, 08:17 AMfareastmovementRe: Analytic problem
- Dec 16th 2011, 08:25 AMFernandoRevillaRe: Analytic problem
- Dec 16th 2011, 08:48 AMfareastmovementRe: Analytic problem
Suppose is analytic on . If for on this region then on the region.

but . How can I make sure that ? Or I am in the wrong direction? Thank you (Itwasntme) - Dec 16th 2011, 12:34 PMFernandoRevillaRe: Analytic problem
Better use this version:

*Let be a bounded domain, and let be a continuous function on the closed set that is analytic on . Then the maximum value of on (which always exists) occurs on the boundary*. So, in our case, it is not possible if .