Bounding $|f'(0)|$ for a complex function $f$ [duplicate]
If $|z|<1$, let$$g(z)=\frac{f(z)-1/2}{1-f(z)/2}.$$Then $g(0)=0$. On the other hand, $g=\varphi\circ f$, with$$\varphi(z)=\frac{z-1/2}{1-z/2}.$$Then, by the Schwarz lemma, $|g'(0)|\leqslant1$. But$$g'(0)=\varphi'(f(0))f'(0)=\varphi'\left(\frac12\right)f'(0)=\frac43f'(0).$$Therefore, $\bigl|f'(0)\bigr|\leqslant\frac34$.