Let $\Omega = \{ z \in \mathbb{C} | Im(z) > 0 \}$ and $f : \Omega \to \mathbb{C}$ a continuous function.
We suppose that $\forall z \in \Omega : |f(z)| \leq M|z|^n$
We define $\gamma : [0,\pi] \to \mathbb{C}$ by $\gamma (t) = Re^{it}$ with $R >0$.
We want to prove that $|\int_\gamma f(z)e^{iz} dz | \leq M \pi R^n$
I think I managed to prove $|\int_\gamma f(z)e^{iz} dz | \leq M \pi R^{n+1}$ using obvious inequalities. So I got $R^{n+1}$ and not $R^n$.
Can someone tell wich one is true ?