An interpolating family of a univalent analytic function
Let \(\Delta\) be the open unit disk and \(\Delta^*\) the complement of the closed unit disk in the complex plane. Let \(\Sigma\) be the class of holomorphic functions which are univalent in \(\Delta^*\) and with development \[ f(z)=z+a_ 0+a_ 1z^{-1}+...\quad. \] Let \(\Sigma_ k\) denote the subclass of \(\Sigma\) consisting of univalent functions f which admit a k'-quasiconformal extension \(\hat f\) to \(\Delta\) with k'\(\leq k.\) The formula \(f_ t(z)=tf(z/t)\), \(0\leq | t| \leq 1\), is an isotopy connecting f to the identity. The author answers positively a question of Bshouty by proving the following theorem about this isotopy. Theorem. If f is in \(\Sigma_ k\), then for any complex number t with \(| t| <1\), \(f_ t\) is in \(\Sigma_{k| t|}\).
scientific article; zbMATH DE number 4077647
scientific article