(a) Let $\mathbb{D}$ denote the unit disk. Is there an analytic function $f \colon \mathbb{D} \to \mathbb{D}$ with $f(0) = 1/2$ and $f′(0) = 3/4?$ Either find such a function $f$ or explain why it does not exist.
(b) Answer the same question for $f(0) = 1/2$ and $f′(0) = 4/5.$
It seems like I could use the Schwarz lemma, but that is not working out so well. Any suggestions? Thanks.
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.
