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Best approximation of the modulus function

Let $\mathbb{D} \subset \mathbb{C}$ be the unit disc. Let $\phi : (\mathbb{D} \to \mathbb{C}) \to \mathbb{R}$ be the following functional: $$ \phi = f \mapsto \sup_{z \in \mathbb{D}} |f(z) - |z|| $$ ...
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