Abstract
The authors mainly concern the set Uf of c ∈ ℂ such that the power deformation \( z(\frac{{f(z)}} {z})^c \) is univalent in the unit disk |z| < 1 for a given analytic univalent function f(z) = z + a2z2 + … in the unit disk. It is shown that Uf is a compact, polynomially convex subset of the complex plane ℂ unless f is the identity function. In particular, the interior of Uf is simply connected. This fact enables us to apply various versions of the λ-lemma for the holomorphic family \( z(\frac{{f(z)}} {z})^c \) of injections parametrized over the interior of Uf. The necessary or sufficient conditions for Uf to contain 0 or 1 as an interior point are also given.
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