Coefficient estimates and radii problems for certain classes of polyharmonic mappings

Abstract

We give coefficient estimates for a class of close-to-convex harmonic mappings F , and discuss the Fekete-Szegő problem of it. We also determine a disk |z| < r in which the partial sum s m,n ( f ) is close-to-convex for each f ∈ F . Then, we introduce two classes of polyharmonic mappings HS p and HC p , consider the starlikeness and convexity of them and obtain coefficient estimates for them. Finally, we give a necessary condition for a mapping F to be in the class HC p .

On a subclass of close-to-convex harmonic mappings

“…Recently, this result is extended in [20] by proving that M(α, −1/2) ⊂ C 0 H for |α| = 1, furthermore, the authors conjectured that M(1, −1/2) ⊂ S 0, * H , however, Nagpal and Ravichandan [21] proved this conjecture is false by giving a counter-example, and they are established some additional results of the class M(1, −1/2). In [13], Chen et al obtained some results of the class M(α, −1/2) with |α| = 1. We refer to [22][23][24][25][26][27][28] for discussions on close-to-convex harmonic mappings.…”

“…Therefore, it is natural to consider the radius of univalence r n of s m,n (f ) (f ∈ S H ). For further results on topic, the reader is referred to [13][14][15][16].…”

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.…”

On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings

“…For special values of α, many authors have studied the class of close-to-convex harmonic mappings, see e.g. [5,9,29,30,39].…”