$f$ be an analytic function defined on $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$

I came across the following problem that says:

Let $f$ be an analytic function defined on $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ such that the range of $f$ is contained in the set $\mathbb{C}\setminus (-\infty,0]$. Then

1. $f$ is necessarily a constant function.
2. there exists an analytic function $g$ on $\mathbb{D}$ such that $g(x)$ is a square root of $f(z)$ for each $z\in\mathbb{D}$.
3. there exists an analytic function $g$ on $\mathbb{D}$ such that $\operatorname{Re}g(z)\geq 0$ and $g(z)$ is a square root of $f(z)$ for each $z\in\mathbb{D}$.
4. there exists an analytic function $g$ on $\mathbb{D}$ such that $\operatorname{Re}g(z)\leq 0$ and $g(z)$ is a square root of $f(z)$ for each $z\in\mathbb{D}$.

I have to determine which options are correct. Can someone help in the right direction? Thanks in advance for your time.

Solution 1:

Option $2,3,4 $ are true, Hint: In a simply connected domain, $f\neq 0$ and analytic, then it is possible to define single valued analytic branches of $\,\,\log f(z)$ and $\displaystyle (f(z))^{\frac{1}{n}}$