The authors mainly concern the set Uf of c E C such that the power deformation z（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...The authors mainly concern the set Uf of c E C such that the power deformation z（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 X-lemma for the holomorphic family z（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.展开更多
基金Project supported by Yeungnam University(2011)(No.211A380226)the JSPS Grant-in-Aid forScientific Research(B)(No.22340025)
文摘The authors mainly concern the set Uf of c E C such that the power deformation z（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 X-lemma for the holomorphic family z（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.
