Variability regions for certain families of harmonic univalent mappings

• Published in: Complex variables and elliptic equations Vol. 58; no. 1; pp. 23 - 34
• Main Authors: Ponnusamy, S., Yamamoto, H., Yanagihara, H.
• Format: Journal Article
• Language: English
• Published: Colchester Taylor & Francis Group 01-01-2013; Taylor & Francis Ltd
• Subjects: analytic; Analytic functions; Complex variables; Disks; harmonic mappings; Harmonics; Mapping; Mathematical analysis; Poisson kernels; Primary: 30C65, 30C45; Schwarz lemma; Secondary: 30C20; starlike; Studies; univalent; variability region
• ISSN: 1747-6933; 1747-6941
• DOI: 10.1080/17476933.2010.551200

Let be the class of analytic functions h in the unit disk . Fix h ∈  with h(0) = h′(0) − 1 = 0 satisfying Re h′(z) > 0 in . For b ∈  , we consider the sets
For fixed z
0
 ∈   \ {0} and b ∈  , we study in this article the regions of variabilities of the sets V
0
(z
0
, b) = {g(z
0
) : g ∈ ℬ(h, b)} and V
1
(z
0
, b) = {g′(z
0
) : g ∈ ℬ(h, b)}. The problem originated from the class of harmonic univalent mappings of the unit disk.