# Variant of Schwarz-Pick for Different Bound/Disk

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?