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Suppose $f$ is analytic in in $D_r(z_0)$ \ {$z_0$}. By Riemman's theorem we know that if $f$ is bounded in some punctured sub disk centered at $z_0$ then $f$ has a removable singularity at $z_0$. Now my question what if all we know is that $Ref(z)$ is bounded in some punctured sub disk centered at $z_0$. Still does it imply that $f$ has a removable singularity at $z_0$?

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    $\begingroup$ Hint: Consider $e^{f(z)}$. $\endgroup$ Dec 4 '14 at 12:40
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Yes. If $\operatorname{Re} f(z) < M$ in the punctured disc then you can apply Riemann's theorem to $g = T \circ f$ where $T$ is a Möbius transformation mapping the half plane $ \{ \operatorname{Re} w < M \} $ onto the unit disk.

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