Abstract
In this paper, our interest is devoted to study the convex combinations of the form \((1-\lambda)f+\lambda g\) where \(\lambda \in(0,1)\) of biholomorphic mappings on the Euclidean unit ball \({\mathbb B}^n\) the case of several complex variables. Starting from a result proved by S. Trimble [26] and then extended by P.N. Chichra and R. Singh [3, Th. 2] which says that if f is starlike such that \(Re[f'(z)]>0\) then \((1-\lambda)z+\lambda f(z)\) is also starlike, we are interested to extend this result to higher dimensions.
In the first part of the paper, we construct starlike convex combinations using the identity mapping on \({\mathbb B}^n\) and some particular starlike mappings on \({\mathbb B}^n\).
In the second part of the paper, we define the class \(L_{\lambda}^∗B^n\) and prove results involving convex combinations of normalized locally biholomorphic mappings and Loewner chains. Finally, we propose a conjecture that generalize the result proved by Chichra and Singh.
Authors
Eduard Stefan Grigoriciuc
Babes-Bolyai” University, Faculty of Mathematics and Computer Science, Cluj-Napoca, Romania
”Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania
Keywords
Biholomorphic mappings; Convex sums; Starlike mappings; Herglotz vector field; Loewner chains
Paper coordinates
E.S. Grigoriciuc, On some convex combinations of biholomorphic mappings in several complex variables, Filomat 36 (16) (2022), 5503-5519;
https://doi.org/10.2298/FIL226503G
About this paper
Journal
Filomat
Publisher Name
Faculty of Sciences and Mathematics, University of Nis, Serbia
Print ISSN
2406-0933
Online ISSN
google scholar link
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