The roles played by order of starlikeness and the Bloch condition in the Roper-Suffridge extension operator
Let \(U\) denote the unit disk in the complex plane, and let \(\mathbb{B}^n\) denote the unit ball in the \(n\)-dimensional complex space \(\mathbb{C}^n\). For \(z=(z_1,\dots,z_n)\in \mathbb{B}^n\), we write \(z=(z_1,z^\prime)\) where \(z^\prime =(z_2,\dots,z_n)\). The Roper-Suffridge extension operator is the operator which sends a normalized locally biholomorphic function \(f\) on the unit disk, to a function on the unit ball \(\mathbb{B}^n\): \[ \Phi_n(f)(z)=(f(z_1),\sqrt{f^\prime(z_1)}z^\prime),\quad z\in \mathbb{B}^n. \] This operator was introduced in 1995 by \textit{K. A. Roper} and \textit{T. J. Suffridge} [J. Anal. Math. 65, 333--347 (1995; Zbl 0846.32006)]. It turned out that the operator \(\Phi\) enjoys several interesting properties: if \(f\) is convex on \(U\), then \(\Phi(f)\) is convex on \(\mathbb{B}^n\); also \(\Phi\) preserves the class of star-like functions as well as the class of Bloch functions. These nice properties motivate mathematicians to study extensions of the Roper-Suffridge operator. For instance, the operator \[ \Phi_G(f)(z)=(f(z_1)+G(\sqrt{f^\prime(z_1)}z^\prime),\sqrt{f^\prime(z_1)}z^\prime),\quad z\in \mathbb{B}^n, \] was introduced by \textit{J. R. Muir} [Complex Anal. Oper. Theory 6, No. 6, 1167-1187 (2012; Zbl 1275.32017)], where \(G:\mathbb{C}^{n-1}\to \mathbb{C}\) is a holomorphic function satisfying specific properties. In the paper under review, the authors study \[ \Phi_{G,p}(f)(z)=(f(z_1)+(G(f^\prime(z_1))^{1/p}z^\prime),(f^\prime(z_1))^{1/p}z^\prime),\quad z\in \mathbb{B}^n, \] for an integer \(p\geq 2\). They prove that if for a certain \(f\), the mapping \(\Phi_{G,p}(f)\) is star-like, then \(G\) is a polynomial of degree at most \(p\).
scientific article; zbMATH DE number 6838046
scientific article