Loewner chains applied to $g$-starlike mappings of complex order of complex Banach spaces

Xiaofei Zhang, Shuxia Feng, Taishun Liu and Jianfei Wang

Vol. 323 (2023), No. 2, 401–431

DOI: 10.2140/pjm.2023.323.401

Abstract

This paper is devoted to studying geometric and analytic properties of $g$-starlike mappings of complex order $\lambda$. By using Loewner chains, we obtain the growth theorems for $g$-starlike mappings of complex order $\lambda$ on the unit ball in reflexive complex Banach spaces, which generalize some results of Graham, Hamada and Kohr. As applications, several different kinds of distortion theorems for $g$-starlike mappings of complex order $\lambda$ are obtained. Finally, we prove that the Roper–Suffridge extension operators preserve the property of $g$-starlike mappings of complex order $\lambda$ in complex Banach spaces, which generalizes many classical results.

Keywords

growth theorems, distortion theorems, modified Roper–Suffridge operator

Mathematical Subject Classification

Primary: 32H02

Secondary: 30C55

Milestones

Received: 17 June 2022

Revised: 14 March 2023

Accepted: 25 March 2023

Published: 2 June 2023