Two theorems in tandem — a bounded-real-part trick and equality in Schwarz–Pick

Consider the statements: (i) An entire function $f$ with $\Re f\le\Im f$ is constant. (ii) There exists an analytic function $f$ from the unit disk $\Delta$ into $\Delta$ such that $f(0)=1/2$ and $f'(0)=3/4$.