Taiwanese Journal of Mathematics

Extension Operators Preserving Janowski Classes of Univalent Functions

Andra Manu

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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 $S^*$).

Article information

Source
Taiwanese J. Math., Volume 24, Number 1 (2020), 97-117.

Dates
Received: 15 May 2018
Revised: 12 March 2019
Accepted: 14 April 2019
First available in Project Euclid: 3 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1556848821

Digital Object Identifier
doi:10.11650/tjm/190407

Mathematical Reviews number (MathSciNet)
MR4053840

Zentralblatt MATH identifier
07175542

Subjects
Primary: 32H99: None of the above, but in this section
Secondary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)

Keywords
$g$-Loewner chain $g$-parametric representation $g$-starlikeness $g$-spirallikeness Janowski starlikeness Janowski almost starlikeness Roper-Suffridge extension operator

Citation

Manu, Andra. Extension Operators Preserving Janowski Classes of Univalent Functions. Taiwanese J. Math. 24 (2020), no. 1, 97--117. doi:10.11650/tjm/190407. https://projecteuclid.org/euclid.twjm/1556848821


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