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February, 2020 Extension Operators Preserving Janowski Classes of Univalent Functions
Andra Manu
Taiwanese J. Math. 24(1): 97-117 (February, 2020). DOI: 10.11650/tjm/190407


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 $S^*$).


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Andra Manu. "Extension Operators Preserving Janowski Classes of Univalent Functions." Taiwanese J. Math. 24 (1) 97 - 117, February, 2020.


Received: 15 May 2018; Revised: 12 March 2019; Accepted: 14 April 2019; Published: February, 2020
First available in Project Euclid: 3 May 2019

zbMATH: 07175542
MathSciNet: MR4053840
Digital Object Identifier: 10.11650/tjm/190407

Primary: 32H99
Secondary: 30C45

Keywords: $g$-Loewner chain , $g$-Parametric representation , $g$-spirallikeness , $g$-starlikeness , Janowski almost starlikeness , Janowski starlikeness , roper-Suffridge extension operator

Rights: Copyright © 2020 The Mathematical Society of the Republic of China


Vol.24 • No. 1 • February, 2020
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