Estimate of line integral of O(x^n) function

Question

Let $f$ be an analytic function in some sector in the complex plane behaving as $$f(z)=\mathcal O(z^n)$$ for some $n$ as $z\to\infty$. Can one prove in general that line integrals of $f$ (in this sector) behave as $$\int_{z_0}^z f(w)dw=\mathcal O(z^{n+1})$$ ?

I hope the question is posed clearly enough. Thank you all very much in advance!

Regards, Frank

Answer 1

Integrals are monotone, so since $|f(z)|\le M|z^n|$ we have for a reasonable path $\gamma$ from $z_0$ to $z$:

$$\left|\int_\gamma f(z)\right|\le\int_\gamma |f(z)|\le M\int_\gamma |z^n|\le M'|z|^{n+1}$$