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

The authors mainly concern the set U _f of c ∈ ℂ such that the power deformation z(f(z)/z) ^c is univalent in the unit disk {pipe}z{pipe} < 1 for a given analytic univalent function f(z) = z + a ₂z ²+... in the unit disk. It is shown that U _f is a compact, polynomially convex subset of the complex plane ℂ 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 λ-lemma for the holomorphic family z(f(z)/z) ^c 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.