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Let $ f : \mathbb{C} \rightarrow \mathbb{C} $ such that $ |f(z)| \leq |sin(z)| $ for every $ z \in \mathbb{C}$. Prove that there exist $ C \in \mathbb{C}$ such that $|C| \leq 1$ such that $ f(z) = C sin z $ for every $ z \in \mathbb{C}$. How to proceed?

rafa
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  • Please provide additional context, which ideally explains why the question is relevant to you and the community. Some forms of context include background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc. – Sahiba Arora Jun 19 '20 at 09:53
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    I presume you want $f$ holomorphic? – Angina Seng Jun 19 '20 at 09:54
  • Answered here https://math.stackexchange.com/q/811393, and more generally, here: https://math.stackexchange.com/q/52121. – Martin R Jun 19 '20 at 09:54

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