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- Jan 26, 2012

- 995

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**Problem**: If $f$ is a holomorphic function on the strip $-1<y<1$, $x\in\mathbb{R}$ with \[\left|f(z)\right|\leq A(1+|z|)^{\eta},\quad\eta\text{ a fixed real number}\]

for all $z$ in that strip, show that for each integer $n\geq 0$ there exists $A_n\geq 0$ so that

\[|f^{(n)}(x)|\leq A_n(1+|x|)^{\eta},\quad\text{for all }x\in\mathbb{R}.\]

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**Hint**:

Use Cauchy's inequality.

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