This seems really simple, but since I didn't find it written anywhere I wanted to make sure I'm not crazy and/or get a reference to a standard textbook where this fact appears.
Let $f:\mathbb{C}\to\mathbb{C}$ be analytic on a disc of radius $R>1$ around the origin. Let $\varphi:S^1 \to \mathbb{C}$ be the restriction of $f$ to the unit circle. Then the negative-index Fourier coefficients of $\varphi$ all vanish.
My idea of a proof: write $f(z)=\sum\limits_{n\geq 0} a_n z^n$. The restriction of that series on the circle is the Fourier series of $\varphi$.
comment - I know that the converse of this statement is not true and that's one way to characterize Hardy spaces. My motivation here is just to see that this direction is easy.