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## The function $f$ is constant

Complex Analysis
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.
Demetres
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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 non-constant 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.
S.F.Papadopoulos
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### Re: The function $f$ is constant

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