No abstract

The aim of this paper is to investigate some properties of planar harmonic and biharmonic mappings. First, we use the Schwarz lemma and the improved estimates for the coefficients of planar harmonic mappings to generalize earlier results related to Landau's constants for harmonic and biharmonic mappings. Second, we obtain a new Landau's Theorem for a certain class of biharmonic mappings. At the end, we derive a relationship between the images of the linear connectivity of the unit disk D under the planar harmonic mappings f = h + g and under their corresponding analytic counterparts F = h − g.

We introduce a class of complex-valued biharmonic mappings, denoted by BH 0 φ k ; σ, a, b , together with its subclass TBH 0 φ k ; σ, a, b , and then generalize the discussions in Ali et al. 2010 to the setting of BH 0 φ k ; σ, a, b and TBH 0 φ k ; σ, a, b in a unified way.

In this paper, our main aim is to discuss the properties of harmonic mappings in the unit ball B n . First, we characterize the harmonic Bloch spaces and the little harmonic Bloch spaces from B n to C in terms of weighted Lipschitz functions. Then we prove the existence of a Landau-Bloch constant for a class of vector-valued harmonic Bloch mappings from B n to C n .2010 Mathematics subject classification: primary 30C65; secondary 30C45, 30C20.

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 .