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Consider $\ f $ be a analytic function defined on D = $\{z \in \mathbb C :|z|<1\}$ such that the range of $\ f$ is contained in the set $\mathbb C$ \ $(- \infty, 0]$. Then there exist an analytic function g on D such that Re g(z) $\geq 0$ and g(z) is a square root of f(z) for each z in D.

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Since $f$ is zero free analytic function in the unit disc, we can define a analytic logarithm of $f(z)$. We define $g(z) = \sqrt{f(z)} = e^{1/2 Log f(z)}$ where $Log f(z)$ is the principal branch of the logarithm that is with branch cut on the negative real axis.

Note that $Re\{g(z)\} = e^{\frac{1}{2} \log{|f(z)|}} e^{i\frac{1}{2}\arg{f}}$.

Since $e^{\frac{1}{2} \log{|f(z)|}} > 0 $ and $-\pi< \arg{f} < \pi \implies \cos{\frac{\arg{f}}{2}} > 0$, hence $Re\{g(z)\} > 0$. I do not think that equality is possible under the given conditions.

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