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let $f$ be an analytic function on a unit disk $D=\left \{ z \in \mathbb{C}: |z|<1 \right \}$ such that the range of the function is contained in the $\mathbb{C} - (-\infty,0]$. Then i have to prove that .

There exist an analytic function $g$ on $D$ such that $g(z)$ is a square root of $f(z)$ for all $z \in D$

I tried my best but not getting even close to this question's approch .How can i find a function which is analytic and range of which function is $\mathbb{C} - (-\infty,0]$ . I think A bilinear transformation can help me because by using that i can map a disk over a disk in .$f$ be an analytic function defined on $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ .The solution is provide here but this solution is talking about analytic branches i Have no idea about them .Please Help.

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  • $\begingroup$ F or this you need the fact that there is an analytic function $h$ on $\mathbb C \setminus (-\infty, 0]$ such that $e^{h(z)}=z$ for all $z$. Once you kn ow this you can take $g(z)=e^{1/2 h(f(z))}$. $\endgroup$ Oct 18, 2019 at 12:57
  • $\begingroup$ In particular note that the shape of $D$ is completely irrelevant $\endgroup$ Oct 18, 2019 at 13:07

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Take $g(z)=\exp\left(\frac{\log f(z)}2\right)$, where $\log$ is an analytic branch of the logarithm defined on $\mathbb C\setminus(-\infty,0]$.

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  • $\begingroup$ You mean $f(z)$, not $z$ $\endgroup$ Oct 18, 2019 at 13:06
  • $\begingroup$ I've edited my answer. Thank you. $\endgroup$ Oct 18, 2019 at 13:07

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