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$f$ is an entire function with $\operatorname{Im}f \geq 0$. Then which of the followings are true: 1) $f$ is constant. 2) $\operatorname{Re}f$ is constant. 3) $f = 0$. 4) $f'$ is a non-zero constant.

That (3) & (4) are wrong can be shown by using $f(z) = i$. But I'm clueless about the remaining two options.

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    I think I've got the point. Non-constant entire function comes arbitrarily close to each complex number. Then both 1 & 2 must be true.2012-11-24

3 Answers 3

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Such a function $f$ would map the entire plane into the upper half-plane. The Picard theorem says this is impossible, since an entire nonconstant function must map the plane onto itself or onto the complex plane punctured by a "missing" point. The mapping $f$ must be constant.

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Hints:
(1) If $f$ is entire then so is $e^{if}$
(2) $|e^{if}|=e^{-\operatorname{Im}(f)}\leq 1$
(3) Use Liouville's theorem

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Consider $g=\dfrac{1}{f+i}$(thanks to point out my mistake), then $g$ is entire and $|g|<1$, now you can conclude that $g$ is a constant with the Liouville's theorem(I forget it's name). Consequently $f$ is also a constant. Also by Picard's theorem, that the range of any nonconstant entire function must contain the complex except for at most one point.

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    You rather mean $g=1/(f+i)$.2012-11-24