Problem statement:
Suppose $V ⊂ \mathbb{C}$ is open and connected such that $\mathbb{D}$ (the unit ball centered at $0$) is contained in $V.$ Let $f : V → \mathbb{C}$ is holomorphic, and $ \int_{\delta \mathbb{D}}f(z) \bar{z}^n =0$ for all $n \in \mathbb{N}.$ Show that $f$ is identically zero on $D.$
This is a very standard exercise. Since holomorphic functions are analytic, I used power series for $f$ at $0.$ Basically, I applied induction on the coefficients $c_n$ of the power series of $f.$ Then I have proved that $c_n=0$ for all $n.$ Is there any other way, this can be proved without induction? Thank you very much.
Although it's quite similar, one can avoid induction by noting that since $\bar z=\frac1z$ on $\delta \mathbb{D}$, $$ \int_{\delta \mathbb{D}}f(z) \bar{z}^n \,dz = \int_{\delta \mathbb{D}} \frac{f(z)}{z^n} \,dz = 2\pi i c_{n-1}. $$