Implications from $f(z)\in\mathbb{R} \Longleftrightarrow z\in \mathbb{R}$

Question

Let $f:D(0,1)\longrightarrow \mathbb{C}$ be a holomorphic function such that $f(z)\in\mathbb{R} \Longleftrightarrow z\in \mathbb{R}$. How to prove that $f$ has at most one zero on the disk.

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By hypothesis the zeros of $f(z)$ are real and $\displaystyle f(z)=\sum_{n=0}^\infty a_nz^n$ , $a_n\in\mathbb{R}$ since $f(z)=\overline{f(\overline{z})}$.

Any help would be appreciated.