The Extension Operators on B n+1 and Bounded Complete Reinhardt Domains
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
THE EXTENSION OPERATORS ON B n+1 AND BOUNDED COMPLETE REINHARDT DOMAINS∗
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Yanyan CUI (
†
)
Chaojun WANG (
College of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China E-mail : [email protected]; [email protected]
4Ó)
Hao LIU (
Institute of Contemporary Mathematics, Henan University, Kaifeng 475001, China E-mail : [email protected] Abstract In this article, we extend the well-known Roper-Suffridge operator on B n+1 and bounded complete Reinhardt domains in Cn+1 , then we investigate the properties of the generalized operators. Applying the Loewner theory, we obtain the mappings constructed by the generalized operators that have parametric representation on B n+1 . In addition, by using the geometric characteristics and the parametric representation of subclasses of spirallike mappings, we conclude that the extended operators preserve the geometric properties of several subclasses of spirallike mappings on B n+1 and bounded complete Reinhardt domains in Cn+1 . The conclusions provide new approaches to construct mappings with special geometric properties in Cn+1 . Key words
Roper-Suffridge operator; Reinhardt domains; spirallike mappings
2010 MR Subject Classification
1
32A30; 30C45
Introduction
There are many graceful conclusions in the theory of one complex variable. To achieve the generalization of the conclusions in spaces of several complex variables, many people began to research biholomorphic mappings with particular geometric properties in Cn , such as starlike mappings, convex mappings and spirallike mappings. Until now, there have been lots of subclasses of these mappings. Let S ∗ (B n ) and K(B n ) denote the classes of normalized starlike and convex mappings on the unit ball B n in Cn . The well-known Roper-Suffridge extension operator ′ q ′ Φn (f )(z) = f (z1 ), f (z1 )z0 ∗ Received March 11, 2019; revised March 27, 2020. This work was supported by NSF of China (11271359, 11471098), Science and Technology Research Projects of Henan Provincial Education Department (19B110016), Scientific Research Innovation Fund Project of Zhoukou Normal University (ZKNUA201805), Scientific Research Fund of High Level Talents of Zhoukou Normal University (ZKNUC2019004). † Corresponding author: Yanyan CUI.
1272
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
was introduced in [1], where z = (z1 , z0 ) ∈ B n , z1 ∈ D, z0 = (z2 , · · ·, zn ) ∈ Cn−1 , f (z1 ) ∈ H(D) p and the branch of the square root is chosen such that f ′ (0) = 1. The operator gives a way of extending a univalent analytic function on the unit disc D in C to a biholomorphic mapping on B n ⊂ Cn . Roper and Suffridge [1] proved that Φn (K) ⊆ K(B n ). Graham and Kohr [2] proved that the Roper-Suffridge operator preserves the properties of Bloch mappings on B n and Φn (S ∗ ) ⊆ S ∗ (B n ), and generalized the Roper-Suffridge operator to be f (z ) β ′ 1 Φn,β (f )(z) = f (z1 ), z0 , z1 1) β where β ∈ [0, 1], (
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