I was trying to construct a holomorphic function $f$ on $\mathbb{C}$ such that $|f|^2(z)=e^{(|z|^2-\frac{1}{2})^2}$. I will be happy if someone can give me an idea how to do that. I would like also to see the function explicity.
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4$\begingroup$ There is no non-linear entire function which has constant modulus even on one circle. $\endgroup$– Alexandre EremenkoCommented May 17, 2022 at 3:43
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$\begingroup$ Do you have a response to the answer below? $\endgroup$– Iosif PinelisCommented May 19, 2022 at 18:04
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1 Answer
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Such a function does not exist, because the constant value $1$ of $|f(z)|^2$ on the circle $\{z\in\mathbb C\colon|z|^2=1/2\}$ is less than the value $e^{1/4}$ of $|f(z)|^2$ at the center $z=0$ of the circle, which would contradict the maximum modulus principle if $f$ were holomorphic on $\mathbb C$.
answered May 16, 2022 at 21:41