# Find an upper bound for nth derivative of a holomorphic function

I was self studying Gamelin's Complex Analysis and came across the following question.

Function $$f(z)$$ is holomorphic in $$\{\operatorname{Im}(z)>0\}$$, i.e., the upper half plane, and bounded by a number $$M$$. Find an upper bound for $$f^{(n)}(z)$$ in $$\{\operatorname{Im}>r\}$$, where $$r$$ is a given positive number.

I think maybe I need to use Cauchy estimates, but Cauchy estimates is only for an open disk situation. Here, it's an open rectangular region. How should I approach this question?

Thanks for any insightful help.

If $$\operatorname{Im} z > r$$ then the disc $$B_r(z)$$ with center $$z$$ and radius $$r$$ is contained in the upper half-plane. The Cauchy estimate can be applied to $$f$$ in this disk, giving $$|f^{(n)}(z) | \le \frac{n! M}{r^n} \, .$$
The map $$\displaystyle g: \mathbb{H} \rightarrow \mathbb{D},\ z \mapsto -i\frac{z+1}{z-1}$$, sends the upper half-plane to the unit disk. Then, $$\displaystyle f(g(z))$$ is on the unit disk and is still bounded above by some $$M$$. Now you can apply Cauchy's estimate on the $$n$$-th derivative of $$f$$.