Let f be a holomorphic function on the strip \(- 1 < y < 1 , x \in \mathbb { R }\) such that
\[ | f ( z ) | \leq A ( 1 + | z | ) ^ { \eta } , \quad \eta \mathrm { ~ i s ~ a ~ f x e d ~ r e a l ~ n u m b e r } \]
for all z in the strip. Show that for each integer \(n \geq 0\) , there exists \(A _ { n }\) such that
\[ | f ^ { ( n ) } ( x ) | \leq A _ { n } ( 1 + | x | ) ^ { \eta } , \quad { \mathrm { ~ f o r ~ a l l ~ } } x \in \mathbb { R } . \]
[Hint: Use Cauchy inequalities]