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Suppose $f \colon \mathbb{C} \to \mathbb{C}$ is an entire function such that $f(0) = 0, f(1) = 1$ and $\vert f(z) \vert \leq 1$ if $\vert z \vert \leq 1$. I want to show that then $f'(1)$ is real and $f'(1) \geq 1$. The Schwarz lemma comes to mind, but it doesn't seem to be of any help here. I tried showing $f'(1) = \overline{f'(1)}$ by considering $g(z) = \overline{g(\overline{z})}$, but this didn't help either. I'm not really sure how to use the hypotheses here. Any hints?

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