In the text "Function Theory of One Complex Variable" Third Edition by Robert E. Greene and Steven G. Krantz i'm inquring if my proof to $\text{Proposition (1)}$ is valid ?
$\text{Proposition (1)}$
If $f$ is a holomorphic polynomial and if
$$\oint_{\partial D(0,1)}f(z) \overline z^{j} dz = 0, \, \, j = 0, 1, 2,.. $$
then prove that $f \equiv 0 $
To begin our quest to prove $(1)$, we write our choice of $f$ as,
$$p(z) = a_{n}(z-a_{1})…(z-a_{n}) = a_{n} \prod_{j}^{n}(z-a_{j}). \tag{1.2}$$
Putting everything together we have that,
$$\oint_{\partial D(0,1)} a_{n}\prod_{j}^{n}(z-a_{j}) \overline z^{j} dz = 0. \tag{1.3}$$
Using Feynman's Integration Trick and the Product rule finally,
\begin{align*} \partial_{\overline z} \bigg( \oint_{\partial D(0,1)} a_{n}\prod_{j}^{n}(z-a_{j}) \overline z^{j} dz \bigg) \tag{1.4} &= \\ \oint_{\partial D(0,1)} \bigg( \partial_{\overline z} a_{n} \prod_{j}^{n}(z-a_{j}) \overline z^{j} dz \bigg) &= 0 \tag{1.5} \end{align*}