Consider $I_\varepsilon :=\oint_{C_\varepsilon} z^αf(z)\,dz,$ where $\alpha>−1$ is real, where $C_\varepsilon$ is a circle of radius $\varepsilon$ centered at the origin and $f(z)$ is analytic inside the circle.
Show that $\lim_{\varepsilon\to 0} I_\varepsilon =0$.
My attempt:
Since $f$ is analytic on the disk, $|f|$ is bounded---and indeed takes its maximum value on the circle (and not in the interior unless it is constant).
Note that using the parametrization $z = \varepsilon e^{i\theta}$, you get $\oint_{C_\varepsilon} \frac{dz}{z} = 2\pi i.$
Okay, so consider writing $z^\alpha = z^{1 + \alpha}/z$. Since $\alpha > -1$, $1 + \alpha > 0$. For fixed $\varepsilon > 0$, let $M(\varepsilon) = \max\{|f(z)| : |z| = \varepsilon\}.$
By the maximum modulus theorem, $M(\varepsilon) ≤ M(\delta)$ whenever $\varepsilon \leq \delta$ ---that is, $M$ is non-decreasing. Now, observe that (all integrals are over $C_\varepsilon$) $ \left|\oint z^\alpha f(z)\,dz\right|\leq \oint|z^{1+\alpha}f(z)|\,\frac{d|z|}{|z|}\leq \varepsilon^{(1+\alpha)}M(\varepsilon)2\pi.$
Since $2\pi M(\varepsilon)$ is bounded and $1 + \alpha > 0$, you can take the limit as $\varepsilon\to 0$ and obtain the desired result.
My question Could it be proved without using the maximum modulus theorem?