Analytic function $f$ in $\overline{\mathbb{D}}$ satisfying $\left\lvert\,f'(\tfrac{1}{2})\,\right \rvert\leq 8.$
Solution 1:
Cauchy's Integral Formula provides that $$ f'\Big(\frac{1}{2}\Big)=\frac{1}{2\pi i}\int_{|z|=1}\!\frac{f(z)\,dz}{\big(z-\frac{1}{2}\!\big)^2}, $$ and hence \begin{align} \left\lvert\, f'\Big(\frac{1}{2}\Big)\right|&\le \frac{\max_{|z|=1}\lvert\,f(z)|}{(1/2)^2}=4\max_{|z|=1}\lvert \,f(z)|\le 4\max_{|z|=1}\big(\lvert\, f(z)-z|+|z|\big) \\&\le 4\max_{|z|=1}\big(|z|+|z|\big)=8. \end{align}