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The function $f$ is constant
 Tolaso J Kos
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The function $f$ is constant
Let \( f \) be an entire function across the complex plane. If \( \mathfrak{Im}(f(z))>\mathfrak{Re}^2 (f(z))2 \) holds, then prove that \( f \) is constant.
Imagination is much more important than knowledge.

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Re: The function $f$ is constant
This is immediate by the little Picard theorem which says that the range of every nonconstant entire function is either the whole of the complex plane, or the complex plane minus one point.
Here however the range of $f$ does not contain any point with imaginary value less than or equal to $2$. Since $f$ is entire, the only possibility left is that $f$ is constant.
Here however the range of $f$ does not contain any point with imaginary value less than or equal to $2$. Since $f$ is entire, the only possibility left is that $f$ is constant.

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Re: The function $f$ is constant
Is easy to prove:
The range of entire no constant function is dense.
The range of entire no constant function is dense.