Suppose $f$ is a holomorphic function on the open unit disk $\mathbb{D}$ with $f(0)=0$ and $| f(z) + zf^{'}(z)| <1$ for all $z \in \mathbb{D}$. I have to show that $|f(z)| \leq \frac{|z|}{2}$ for all $z\in \mathbb{D}$.
I have tried to apply Schwarz Lemma but failed to obtain the inequality.