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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.

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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 $$ |f^{(n)}(z) | \le \frac{n! M}{r^n} \, . $$

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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 $f$.

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