Coefficient Estimates and Bloch’s Constant in Some Classes of Harmonic Mappings

Abstract

Following Clunie and Sheil-Small, the class of normalized univalent harmonic mappings in the unit disk is denoted by S H. The aim of the paper is to study the properties of a subclass of S H , such that the analytic part is a convex function. We establish estimates of some functionals and bounds of the Bloch's constant for co-analytic part.

Some classes of harmonic mappings with analytic part defined by subordination

“…We provide coefficient estimates, distortion, and area estimates, and covering theorems of the classes. The results presented here generalize the main results in [9,13,14,32]. Now we introduce the following classes.…”

“…In [15], the authors studied the properties of a subclassS α H of S H , consisting of all univalent antianalytic perturbations of the identity in the unit disk with |b 1 | = α , and in [16], the authors studied the classS α H of all f ∈ S H , such that |b 1 | = α ∈ (0, 1) and h ∈ CV , where CV denotes the well-known family of normalized, univalent functions that are convex. Recently, Kans and Klimek-Smet [14] studied the properties of a subclass ofS α H and established estimates of some functionals and bounds of the Bloch constant for the coanalytic part. Also, Hotta and Michalski [9] studied the properties of a subclass of L H having starlike analytic part h and obtained coefficient, distortion, and growth estimates of g , and Jacobian estimates of f .…”

“…Also, the function f (z) ∈K α H (A, B) if and only if f ∈ S H , |b 1 | = α ∈ [0, 1) and h(z) ∈ K(A, B). Additionally, we define the classes (Klimek and Michalski [16]; Kans and Klimek-Smet [14]).…”

Convolution Properties of Some Harmonic Mappings in the Right Half-Plane

“…In particular, a domain convex in the horizontal direction is denoted by CHD. For more basic details on univalent harmonic mappings, we refer the reader to [1][2][3] and for some recent results one can see [4][5][6][7].…”

Harmonic Mappings Related to Functions With Bounded Boundary Rotation and Norm of the Pre-Schwarzian Derivative

“…We remain, for example, papers [13], [14], [15]; in [14] authors studied properties of a subsetS α H of S H consisting of all univalent anti-analytic perturbations of the identity whereas in [15] the class S α of all f ∈ S H , such that h is convex, normalized univalent functions.…”