Variant of Schwarz-Pick for Different Bound/Disk

Question

All,

I'm looking to prove an alternate version of the Schwarz-Pick Lemma:

Let $f:D(0,r) \rightarrow \mathbb{C}$ be holomorphic, and suppose that $|f(z)| \leq U \quad \forall z \in D(0,r)$. Then, $\forall z \in D(0,r)$, it holds that

$$ |f'(z)| \leq \frac{r(U^2 - |f(z)|^2)}{U(R^2 - |z|^2)} $$

My initial thought was to consider the Mobius transform $$ M(z) = \frac{z - z_0}{r - \bar z_0z} $$ for $z \in D(0,r)$, and develop a composition that satisfies the Schwarz Lemma, yielding the desired conclusion, but I can't seem to find one that works. Is this claim even true? If so, how should I proceed?

Answer 1

First, prove the Schwarz-Pick Lemma ($U=R=1$). Then apply the following hint:

Hint: how can you write $f$ as a function $g\colon D(0,1)\to D(0,1)$? Apply the Schwarz-Pick Lemma to your $g$.