Roper-Suffridge type extension operators for univalent mappings revisited

Abstract

Let f be a normalized univalent function on the unit disc U, and let $\alpha ,\beta \in R$. We consider a family of operators ${\Psi }_{\alpha ,\beta }$ that extend f to biholomorphic mappings defined on the unit ball B of a complex Hilbert space $H$ into $H$, and they are given by ${\Psi }_{\alpha ,\beta }\left(f\right)\left(z\right)=\left(f\right(z_1),w{\left(\frac{f\left(z_1\right)}{z_1}\right)}^{\alpha }{\left(f^{\prime }\right(z_1\left)\right)}^{\beta })$, where, for a fixed unit vector $e_1\in H$, we use the notation $z=(z_1,w)$ if $z=z_1{e}_1+w$ and w is orthogonal to $e_1$. In the case $\alpha =0$ and $\beta =1/2$ we obtain the Roper-Suffridge extension operator. Until now, it is only known that for the pairs $(\alpha ,\beta )\in R^2$ such that $0\le \alpha \le 1$, $0\le \beta \le 1/2$ and $\alpha +\beta \le 1$, ${\Psi }_{\alpha ,\beta }\left(f\right)$ can be embedded as the initial element of a normal Loewner chain on B for any normalized univalent function f on U. In this paper, we describe a closed domain D in $R^2$ such that for $(\alpha ,\beta )\in D$, ${\Psi }_{\alpha ,\beta }\left(f\right)$ can be embedded as the initial element of a normal Loewner chain on B for any normalized univalent function f on U. The closed domain D contains the above mentioned closed subset defined by $0\le \alpha \le 1$, $0\le \beta \le 1/2$ and $\alpha +\beta \le 1$, and also it contains points $(\alpha ,\beta )$ such that $\alpha <0$ and/or $\beta <0$. For the proof of the main results, we use a new method which is different from the methods developed in the papers by Graham, Hamada, Kohr and Kohr in 2020 and Graham, Hamada, Kohr and Suffridge in 2002. As a corollary, we obtain that for any $(\alpha ,\beta )\in D$, ${\Psi }_{\alpha ,\beta }$ preserves starlikeness, that is, if f is a normalized starlike function on U, then ${\Psi }_{\alpha ,\beta }\left(f\right)$ is a starlike mapping on B. We also show that if $(\alpha ,\beta )\in R^2\setminus \left\{\right(0,1/2\left)\right\}$, then the operator ${\Psi }_{\alpha ,\beta }$ does not preserve convexity.