Is there an analytic function $f : \mathbb{D} → \mathbb{D}$ with $f(0) = 1/2$ and $f′(0) = 3/4?$

The "Schwarz-Pick" variant of Schwarz's lemma, proved, as Greg Martin's comment suggests, by composing with Möbius transformations of the unit disk, exactly answers your question: a holomorphic function $f\colon \mathbb{D} \to \mathbb{D}$ must satisfy

$$\frac{|f'(z)|}{1-|f(z)|^2} \le \frac{1}{1-|z|^2}$$

In case (a), the Schwarz-Pick inequality is saturated for $z=0$, so basically the only possible function is the Möbius transformation $f(z) = \frac{2z-1}{z-2}$ taking $0$ to $\frac{1}{2}$, which has derivative $-\frac{3}{4}$ there. In case (b), the inequality forbids such a function.