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

Question

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 need your help. Thanks.

Answer 1

Since it's homework, a few hints:

First assume that $f$ is holomorphic on a neighbourhood of $\bar D$. Then use the maximum principle in each variable separately to conclude that it's true in this case. Finally, approximate your $f$ by functions that are holomorphic on a neihbourhood of $\bar D$.

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More details I'll do it for $n=2$ under the assumption that $f$ extends to a neighbourhood of $\bar D$, and leave the general case up to you. Let $a \in \partial \Delta$ and define $\phi_a(\zeta) = (a,\zeta)$. Then $f(\phi_a(\zeta))$ is holomorphic on a neighbourhood of $\bar\Delta$, so by the maximum modulus principle in one variable applied to $f\circ \phi_a$, $\sup_{\zeta\in\bar\Delta} f(a, \zeta) = \sup_{\zeta\in\partial\Delta} f(a, \zeta)$. Do the same for $\psi_a(\zeta) = (\zeta,a)$ and take the sup over all $a \in \Delta$ to obtain $$\sup_{z\in \partial(\Delta \times \Delta)} f(z) = \sup_{z\in\partial\Delta\times\partial\Delta} f(z)$$ since the union of the images of $\phi_a$ and $\psi_a$ cover the entire boundary of $\Delta\times\Delta$. To finish off, use the maximum modulus principle in $\mathbb{C}^n$ to see that $$\sup_{z\in \bar\Delta \times \bar\Delta} f(z) = \sup_{z\in\partial(\Delta\times\Delta)} f(z)$$ or, if you prefer, do the same argument again with $a \in \Delta$.

Answer 2

I'm not sure but I still post my solutions for the problem, if I have some mistakes, then you fix it. Thanks.

Case 1: $z\in D$

Since $D$ is bounded, its closure $\overline{D}$ is closed and bounded, hence compact.

$|f(z)|$ is continuous real-valued function on the compact set and so $\sup|f(z)|<\infty$. By the maximum modulus principle, we have $f$ is constant.

Because $f \in \mathcal{C}(\overline{D})$, therefore:$\sup_{z \in \overline{D}} |f(z)|=\sup_{z \in \Gamma} |f(z)|$.

Case 2: $z \in \Gamma$, it's obvious.

Case3: $z \in \partial D \setminus \Gamma$,

We assume that $z_{1}^{0}, \ldots,z_{n}^{0}$ such that: $|z_j-a_j|=r_j, \forall j=1,\ldots, n_0$ and $z_{n_0+1}^{0},\ldots,z_{n}^{0}:|z_j-a_j|<r_j, \forall j=n_0+1,\ldots, n$.

We consider: $g :D' \to \mathbb{C} $, with $D'=\Delta(a_{n_0+1},r_{n_0+1}) \times \ldots \times \Delta(a_{n},r_{n})$ and $g$ is defined by

$z'=(z_{n_0+1}, \ldots , z_n) \mapsto g(z')=f(z_{1}^{0}, \ldots,z_{n}^{0},z_{n_0+1}, \ldots, z_n)$, so $g \in \mathcal{H}(D') \cap \mathcal{C}(\overline{D'})$.

If $g$ has $\sup$ at $z'^0=(z_{n_0+1}^{0},\ldots,z_{n}^{0}) \in D'$. By the maximum modulus principle, we have $g$ is constant in $D'$.

We have:

$\sup_{z\in \overline{D}}|f(z)|=|f(z_0)|=|f(z_{1}^{0}, \ldots,z_{n}^{0})|= |g(z'^0)|=|g(z^0)|$.

Where $z^*=(z_{n_0+1}^{*},\ldots,z_{n}{*}) \in \Gamma'$

$\implies \sup_{z\in \overline{D}}|f(z)|=|f(z_{1}^{0}, \ldots,z_{n}^{0},z_{n_0+1}^{*},\ldots,z_{n}{*})|=\sup_{z \in \Gamma} |f(z)|$.

Is that right? Thanks.