On univalence of the power deformation z(f(z)/z)(c)

Abstract

The authors mainly concern the set U (f) of c a a", such that the power deformation is univalent in the unit disk |z| < 1 for a given analytic univalent function f(z) = z + a (2) z (2) + aEuro broken vertical bar in the unit disk. It is shown that U (f) is a compact, polynomially convex subset of the complex plane a", unless f is the identity function. In particular, the interior of U (f) is simply connected. This fact enables us to apply various versions of the lambda-lemma for the holomorphic family of injections parametrized over the interior of U (f) . The necessary or sufficient conditions for U (f) to contain 0 or 1 as an interior point are also given.