Starlikeness and convexity of polyharmonic mappings

Abstract

In this paper, we first find an estimate for the range of polyharmonic mappings in the class HC 0 p . Then, we obtain two characterizations in terms of the convolution for polyharmonic mappings to be starlike of order α, and convex of order β, respectively. Finally, we study the radii of starlikeness and convexity for polyharmonic mappings, under certain coefficient conditions. 2010 Mathematics Subject Classification. Primary 30C65, 30C45; Secondary 30C20.

Compositions of polyharmonic mappings

“…where each G k (z) is harmonic in D. There is now a long list of articles in the literature on this subject. For recent results on p-harmonic mappings, we refer to the articles [8,9,10,11,12,18,20,21,22,23]. Another motivation for the study of polyharmonic mappings is from the recent work of Borichev and Hedenmalm [5] on the study of second order elliptic partial differential equations T α (f ) = 0 in the unit disk D = {z : |z| < 1}, where α ∈ R and…”

A note on polyharmonic mappings

“…By using the representation (1.1), many properties of harmonic mappings can be generalized to the polyharmonic mappings. This line of research was started in by S. Chen, Ponnusamy, Qiao, and Wang in [4,21], and continued by the third author with J. Chen and Wang in the series of articles [5,6,7,8,9] where, for example, the radii of univalency, starlikeness and convexity were studied for polyharmonic mappings.…”

Geometric Properties of log-Polyharmonic Mappings

“…[2,4,5,13]). The reader is referred to [7,8,10,11,12] for further discussions on polyharmonic mappings and [9,14,16] as well as the references therein for the properties of harmonic mappings.…”