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        general >> complex analysis >> Maximum modulus principle
        
(Message started by: kimtahe6 on Jun 20^th, 2013, 8:06am)

Title: Maximum modulus principle
Post by kimtahe6 on Jun 20^th, 2013, 8:06am

Suppose $D=\Delta^n(a,r)=\Delta(a_1,r_1)\times \ldots \times \Delta(a_n,r_n) \subset \mathbb{C}^n$

and

$\Gamma =\partial \circ \Delta^n(a,r)=\left \{ z=(z_1, \ldots , z_n)\in \mathbb{C}^n:|z_j-a_j|=r_j,~ j=\overline{1,n} \right \}$.

Let $f \in \mathcal{H}(D) \cap \mathcal{C}(\overline{D})$.

Prove that: $\sup_{z \in \overline{D] |f(z)|=\sup_{z \in \Gamma} |f(z)|$

I think we apply maximum modulus principle, but i have trouble...
Any help (or hint or another solution) would be greatly appreciated :-* . Thanks.