Extensions of Abstract Loewner Chains and Spirallikeness

Muir, Jerry R.

doi:10.1007/s12220-022-00921-3

Cited by 2 publications

(1 citation statement)

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“…The extension operator Φ n,Q preserves parametric representation and starlikeness if Q ≤ 1/4 (see [10]), convexity if Q ≤ 1/2 ( see [16]) and starlikeness of order α ∈ (0, 1) if Q ≤ 1−|2α−1| 8α (see [21]; see also [2]). In a recent study, there has been investigated results concerning extended Loewner chains and this extension operator, as well as other preservation results (see [15]). Also, modifications of the Muir extension operator were considered in [6].…”

Section: Preliminariesmentioning

confidence: 99%

Extension operators and Janowski starlikeness with complex coe cients

Manu

2023

Stud. Univ. Babes-Bolyai Math.

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"In this paper, we obtain certain generalizations of some results from [13] and [14]. Let $\Phi_{n, \alpha, \beta}$ be the extension operator introduced in \cite{GrahamHamadaKohrSuffridge} and let $\Phi_{n, Q}$ be the extension operator introduced in [7]. Let $a \in \C$, $b \in \R$ be such that $|1-a| < b \leq {\rm Re}\ a$. We consider the Janowski classes $S^*(a,b, \B)$ and $\A S^*(a,b, \B)$ with complex coefficients introduced in [16]. In the case $n=1$, we denote $S^*(a,b, \mathbb{B}^1)$ by $S^*(a,b)$ and $\A S^*(a,b, \mathbb{B}^1)$ by $\A S^*(a,b)$. We shall prove that the following preservation properties concerning the extension operator $\Phi_{n, \alpha, \beta}$ hold: $\Phi_{n, \alpha, \beta} (S^*(a,b)) \subseteq S^*(a,b, \B)$, $\Phi_{n, \alpha, \beta} (\A S^*(a,b)) \subseteq \A S^*(a,b, \B)$. Also, we prove similar results for the extension operator $\Phi_{n, Q}$: $$\Phi_{n, Q}(S^*(a,b)) \subseteq S^*(a,b, \B),\ \Phi_{n, Q}(\A S^*(a,b)) \subseteq \A S^*(a,b, \B).$$ "

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