# $f$ is analytic function over a unit disk.

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.

• 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 $g(z)=e^{1/2 h(f(z))}$. Oct 18, 2019 at 12:57
• In particular note that the shape of $D$ is completely irrelevant Oct 18, 2019 at 13:07

Take $$g(z)=\exp\left(\frac{\log f(z)}2\right)$$, where $$\log$$ is an analytic branch of the logarithm defined on $$\mathbb C\setminus(-\infty,0]$$.
• You mean $f(z)$, not $z$ Oct 18, 2019 at 13:06