A possible application of Schwarz lemma

Question

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[1]: Let h : C → C be an analytic function such that h(0) = 0; h(1\2) = 5, and |h(z)| < 10 for |z| < 1. The first two options can be eliminated using the function $h(z)=10z$. Third option is true by Maximum Modulus Principle because it says the maximum is obtained only on the boundary. However, I could not prove the last one. Is Schwarz lemma useful here? If yes, how?

Answer 1

Apply Schwarz Lemma to $\frac {f(z)} {10}$. We get $|f(z)| \leq 10|z|$ and, since equality holds for $z=\frac 1 2$, we must have $f(z)=10z$ (by the second half of Schwarz Lemma). Hence c) and d) are true.