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Suppose $f$ is an entire function satisfying $|f(z)|\le|z||x|$ where $x=\text{real part of }z$. Then prove that $f(z)=0$ for every $z\in \mathbb{C}$

I am able to prove that $f(0)=0$ and $f'(0)=0$. I need to prove that all derivatives vanish at 0. How do I prove that? Help please.

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1 Answer 1

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The hypothesis implies that $|f(z)| = 0$ (and hence $f(z) = 0$) for all pure imaginary $z$. Since the imaginary axis is a set with a limit point, it follows by analytic continuation that $f(z) = 0$ for all $z$.

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  • $\begingroup$ Great! Thank you. Identity theorem. Elegant. $\endgroup$ Dec 7, 2020 at 23:52

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