On entire conformal mappings of simply connected regions

Abstract:

Let $\Omega$ be a simply connected proper neighborhood of the origin in the complex plane. Let $\phi$ be a one to one conformal mapping of the unit disk onto $\Omega$, with $\phi (0) = 0$. The most general one to one conformal mapping of $\Omega$ onto itself which fixes the origin has the form ${f_\lambda }(z) = \phi (\lambda {\phi ^{ - 1}}(z))$ where $|\lambda | = 1$. It is shown that the set of $\lambda$ for which ${f_\lambda }$ is entire and nonlinear has Lebesgue measure zero on the unit circle. The proof depends in part upon properties of solutions, $\phi$, of the functional equation $f(\phi (w)) = \phi (\lambda w)$, where $f$ is an entire, nonlinear function, $f(0) = 0$ and $|\lambda | = 1$.