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A complex condition for a function to be 0

Source: Admission to SNSB, 2002

October 4, 2019
functioncomplex analysis

Problem Statement

Let f:D⟶C f:\mathbb{D}\longrightarrow\mathbb{C} be a continuous function, where D \mathbb{D} is the closed unit disk. Suppose that f f is holomorphic on the open unit disk and that eiθ e^{i\theta } are roots, for any θ∈[0,π/4]. \theta\in\left[ 0,\pi /4 \right] . Show that f=0D. f=0_{\mathbb{D}} .
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