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