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Open subset of complex plane

Source: IMS 2014 - Day2 - Problem12

October 5, 2014

complex analysiscomplex analysis unsolved

Problem Statement

Let

U

be an open subset of the complex plane

\mathbb{C}

including

\mathbb{D}=\{z \in \mathbb{C} : |z| \le 1\}

and

f

be analytic over

U

. Prove that if for every

z

with a complex norm equal to

1

(

|z|=1

) we have

0<Re(\bar{z}f(z))

, then

f

has only one root in

\mathbb{D}

and that's simple.

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