Posts by Eduard Grigoriciuc

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

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About this paper

Journal

Computational Methods and Function Theory

Publisher Name

Springer

Print ISSN
1617-9447
Online ISSN

2195-3724

google scholar link

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