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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.

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

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  • $\begingroup$ i think both branch of square root of f will work... $\endgroup$ – neelkanth Jun 2 '15 at 10:25
  • $\begingroup$ First you map the unit disk to the right-hand half-plane by a fractional-linear transformation. Then apply square root. $\endgroup$ – Lubin Jun 3 '15 at 4:22
  • $\begingroup$ If possible please solve it completely....thanks in advance $\endgroup$ – neelkanth Jun 4 '15 at 3:02
  • $\begingroup$ My comment was completely wrong, sorry. I’ll also comment below. $\endgroup$ – Lubin Jun 4 '15 at 17:24
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Clearly both branches of $f(z)^{1/2}$ will form the required function $g.$

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  • $\begingroup$ 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. $\endgroup$ – Lubin Jun 4 '15 at 17:26
  • $\begingroup$ Can you elaborate more on how both the branches of $f(z)^{\frac 12}$ form the required function $g$? $\endgroup$ – Error 404 Jun 15 '18 at 9:23

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