A function $f$ is analytic in $ D=D(0,1)$ and $f(0)=f'(0)=0$

Question

A function $f$ is analytic in $ D=D(0,1)$ and $f(0)=f'(0)=0$ and that $|f'(z)|\leq 1$ for every $z \in D.$ Prove that $|f(z)|\leq |z|^2/2$ for every $z \in D.$

I always have problems to get an inequality for $f$ from $f'.$ I can note this is similar to one version of Schwarz's Lemma: "If $f$ is analytic in the unitary disk $\mathbb D$ and $f(0)=0$ and for every $z \in \mathbb D$ then $|f(z)|\leq |z|.$

Thanks so much!

Answer 1

So $|f'(z)|\leq|z|$, and $|f(z)|=\left|\displaystyle\int_{[0,z]}f'(w)dw\right|\leq|z|^{2}\displaystyle\int_{0}^{1}tdt=\dfrac{|z|^{2}}{2}$.