# Entire function which preserves unit disk and fixes $0$ and $1$

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?