Extension Operators Preserving Janowski Classes of Univalent Functions

Abstract

In this paper, our main interest is devoted to study the extension operator $\Phi_{n,\alpha,\beta} \colon \mathcal{L}S \to \mathcal{L}S_n$ given by $\Phi_{n,\alpha,\beta}(f)(z) = \big( f(z_1), \widetilde{z}(f(z_1)/z_1)^{\alpha} (f'(z_1))^{\beta} \big)$, $z = (z_1,\widetilde{z}) \in \mathbf{B}^n$, where $\alpha,\beta \geq 0$. We shall prove that if $f \in S$ can be embedded as the first element of a $g$-Loewner chain with $g \colon U \to \mathbb{C}$ given by $g(\zeta) = (1+A\zeta)/(1+B\zeta)$, $|\zeta| \lt 1$, and $-1 \leq B \lt A \leq 1$, then $F = \Phi_{n,\alpha,\beta}(f)$ can be embedded as the first element of a $g$-Loewner chain on the unit ball $\mathbf{B}^n$ for $\alpha \in [0,1]$, $\beta \in [0,1/2]$ and $\alpha + \beta \leq 1$. As a consequence, the operator $\Phi_{n,\alpha,\beta}$ preserves the notions of Janowski starlikeness on $\mathbf{B}^n$ and Janowski almost starlikeness on $\mathbf{B}^n$. Particular cases will be also mentioned.

On the other hand, we are also concerned about some radius problems related to the operator $\Phi_{n,\alpha,\beta}$ and the Janowski class $S^*(a,b)$. We compute the radius $S^*(a,b)$ of the class $S$ (respectively $S^*$).