New Subclasses of Univalent Mappings in Several Complex Variables: Extension Operators and Applications

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Abstract

In this paper we define new subclasses of univalent mappings in the case of several complex variables. We will focus our attention on a particular class, denoted $E_{1}^{\ast}$, and observe that in the case of one complex variable, $E_{1}^{\ast}(U)$\ coincides with the class of convex functions $K$ on the unit disc. However, if $n\geq2$, then $E_{1}^{\ast (\mathbb{B}^{n})$ is different from the class of convex mappings $K(\mathbb{B}^{n})$ on the Euclidean unit ball $\mathbb{B}^{n}$ in $\mathbb{C}^{n}$. Along with this, we will study other properties of the class $E_{1}^{\ast}$ on the unit polydisc, respectively on the Euclidean unit ball in $\mathbb{C}^{n}$. In the second part of the paper we discuss the Graham–Kohr extension operator n,\U{3b1} (defined by Graham and Kohr in Complex Variab. Theory Appl. 47:59-72, 2002). They proved that the extension operator $\Psi_{n,\U{3b1} }$ does not preserve convexity for $n\geq2$ for all $\U{3b1} \in \lbrack0,1]$. However, in this paper we prove that $\Psi_{n,0}(K)$ and $\Psi_{n,1}(K)$ are subsets of the class $E_{1}^{\ast}\left( \mathbb{B}^{n}\right)$ which is different from the class $K(\mathbb{B}^{n})$ for the Euclidean case.

Authors

Eduard Ştefan Grigoriciuc
Department of Mathematics, Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania

Keywords

Biholomorphic mapping; Convex mapping; Starlike mapping; Extension operator.

Paper coordinates

E.S. Grigoriciuc, New subclasses of univalent mappings in several complex variables: extension operators and applications, Computational Methods and Function Theory, 23 (2023), pp. 533–555, https://doi.org/10.1007/s40315-022-00467-z

PDF

https://doi.org/10.1007/s40315-022-00467-z

About this paper

Journal

Computational Methods and Function Theory

Publisher Name

Springer

DOI

https://doi.org/10.1007/s40315-022-00467-z

Print ISSN

1617-9447

Online ISSN

2195-3724

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