# Analytic function defined on unit disc with co-domain except negative real axis. [duplicate]

I am trying to solve the following problem. Let $f$ be an analytic function defined on $\mathbb{D}=\{z:|z| <1 \}$ such that range of $f$ is contained in $\mathbb{C}$ \ $(-\infty,0]$.Then there exist an analytic function $g$ such that $Re g(z) \ge 0$ and $g$ is a square root of $f$ on $\mathbb{D}$ and there exist an analytic function $g$ such that $Re g(z) \le0$ and $g$ is a square root of $f$ on $\mathbb{D}$. Clearly fuction $Log(f(z))$ and function $f(z)^{1/n}$are well defined according to given co-domain of $f$ . But these functions are not required function according to me. Then how to find required function $g$. Since $f$ is non zero so $f^{'}/f$ is also analytic and there is a function $g$ such that $g=Log(f(z))$. Now i am stuck please some one help.

## marked as duplicate by Eevee Trainer, user593746, BigbearZzz, user10354138, Paul FrostDec 15 '18 at 23:13

Clearly both branches of $f(z)^{1/2}$ will form the required function $g.$
• I disagree with these comments. Since $f$ maps into the slit disk (plane minus the nonpositive real axis), a domain on which the square root is well defined in such a way that $\sqrt1=1$, OP may indeed take the square root of $f$ to solve his problem. – Lubin Jun 4 '15 at 17:26
• Can you elaborate more on how both the branches of $f(z)^{\frac 12}$ form the required function $g$? – Error 404 Jun 15 '18 at 9:23