= Odd square-root transform of a normalized univalent function
{title2=$h(z)=\sqrt{f(z^2)}=z+\tfrac12a_2z^3+\cdots$}
= Odd square-root transform
{synonym}
Because $f(z)/z$ is holomorphic and nowhere zero on the <unit disc>, its <square root> with value one at zero exists. Defining $h(z)=z\sqrt{f(z^2)/z^2}$ gives an odd normalized <univalent function>. If $h(z)=h(w)$, squaring gives $z^2=w^2$; the possibility $z=-w$ and oddness force both to be zero. Applying the <area theorem for univalent functions> to $1/h(1/\zeta)$ yields $|a_2|\le2$.
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