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Let $R>0$ and $f: K_R(0) \rightarrow \mathbb{C}, f(z) = \sum_{k=0}^{\infty} a_kz^k$ be a compley power series with convergence radius $R>0$ such that $f(K_R(0)) \subset \mathbb{R} $, where $K_R(0)$ is the disk around zero with radius R. I would like to show that all coefficients $a_k$ with $k \in \mathbb{N} $ vanish.

So far I have noticed that if the above statement is true, the power series must value the constant $a_0$ for any $ z \in K_R(0) $ because otherwise the $a_k$ would not have to be zero for all $k>0$. My problem is that I do not see why there could not be any coefficients such that $\sum_{k=1}^{\infty} a_kz^k$ would be real as well.

My idea is to consider the coefficients in the form $a_k = \frac{f^{(k)}(0)}{k} $. If the power series $f(z)$ does value a constant $a_0$, then the derivatives $f^{(k)}=0$ , so that $a_k = 0$ for $k \in \mathbb{N}$.

Am I on the right path? Any hint would be highly appreciated.

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A real valued analytic function on any domain is a constant. [One way to prove this is to use Cauchy-Riemann equations]. Since $f$ is anlytic it follows that $f(z)=k_0$ for all $z$.

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