Suppose $f(z) = z + a_2 z^2 + \cdots$ with $f$ univalent on the unit disc, $f(0) = 0$, and $f'(0) = 1$. Define $g(z)^2 = f(z)/z$ and $h(z) = zg(z^2)$. Then $h$ is univalent on the disc, with $h(0) = 0$ and $h'(0) = 1$. Note that $h$ is odd, and $h(z)^2 = f(z^2)$ for each $z$ in the unit disc. How do you see that $h(z) = z + \frac{a_2}{2} z^2 + \cdots$?
EDIT: Added missing hypotheses.