Univalent Harmonic Mappings Research Articles

Injectivity of sections of univalent harmonic mappings

If S denotes the class of functions h(z)=z+∑n=2∞anzn which are analytic and univalent in the unit disk |z|<1, then the classical result of Szegö shows that every section sn(h)(z)=∑k=1nakzk of h is univalent in |z|<1/4. Exact (largest) radius of univalence rn of sn(h) remains an open problem, although the corresponding results for sections of various geometric subclasses of S have been obtained which include those h∈S for which Reh′(z)>0 holds in the unit disk. However, no attempt has been made to derive harmonic analog of these results. The central object in this case is the class SH0 of sense-preserving harmonic univalent mappings f=h+g¯ defined on the unit disk, normalized so that h(0)=g(0)=h′(0)−1=g′(0)=0. Our primary objective in this paper is to solve the univalency of every section of a harmonic function in the class PH0(α), where PH0(α) denotes the class of normalized univalent harmonic mapping f=h+g¯ in the unit disk |z|<1 satisfying the condition Re(h′(z)−α)>|g′(z)| for |z|<1, where g′(0)=0 and 0≤α<1. In this paper, we first present sharp bounds for the moduli of the coefficients an, bn of f∈PH0(α) and then determine the value of r so that the partial sums of f∈PH0 are close-to-convex in |z|<r.

Polynomial approximation of harmonic mappings

This study is concerned with univalent harmonic mappings on the unit disc of the complex plane. We examine a family of harmonic polynomials that have some interesting geometric properties, e.g. the boundary curve is concave except for cusps, that is dense in the family of harmonic mappings.

Convolutions of slanted half-plane harmonic mappings

Let S 0 ( H γ ) denote the class of all univalent, harmonic, sense-preserving and normalized mappings f of the unit disk D onto the slanted half-plane H γ : = { w :Re( e iγ w ) &gt; −1/2} with an additional condition f z (0) = 0. Functions in this class can be constructed by the shear construction due to Clunie and Sheil-Small which allows by examining their conformal counterpart. Unlike the conformal case, convolution of two univalent harmonic convex mappings in D is not necessarily even univalent in D. In this paper, we fix f 0 ∈ S 0 ( H 0 ) and show that the convolutions of f 0 and some slanted half-plane harmonic mapping are still convex in a particular direction. The results of the paper enhance the interest among harmonic mappings and, in particular, solve a recent open problem of Dorff et al. in a more general setting. Finally, we present some basic examples of functions and their corresponding convolution functions with specified dilatations, and illustrate them graphically with the help of MATHEMATICA software. These examples explain the behaviour of the image domains.

Pre-Schwarzian and Schwarzian Derivatives of Harmonic Mappings

In this paper we introduce a definition of the pre-Schwarzian and the Schwarzian derivatives of any locally univalent harmonic mapping f in the complex plane without assuming any additional condition on the (second complex) dilatation ω f of f. Using the new definition for the Schwarzian derivative of harmonic mappings, we prove theorems analogous to those by Chuaqui, Duren, and Osgood. Also, we obtain a Becker-type criterion for the univalence of harmonic mappings.

Subclasses of Harmonic Mappings Defined by Convolution

Two new subclasses of harmonic univalent functions defined by using convolution and integral convolution are introduced. These subclasses generate several known and new subclasses of harmonic univalent functions as special cases and provide a unified treatment in the study of these classes. Coefficient bounds, extreme points, distortion bounds, convolution conditions, and convex combination are also determined.

Some properties of generalized convolution of harmonic univalent functions

Abstract The purpose of the present paper is to investigate some interesting properties on generalized convolutions of functions for the classes HP ∗(α), HS(α) and HC(α). Further, an application of the convolution on certain integral operator are mentioned.

Fully starlike and fully convex harmonic mappings of order α

The hereditary properties of convexity and starlikeness for conformal mappings do not generalize to univalent harmonic mappings. This failure leads to the notions of fully starlike and fully convex mappings. In this paper, properties of fully starlike

Variability regions for certain families of harmonic univalent mappings

Let 𝒜 be the class of analytic functions h in the unit disk 𝔻. Fix h ∈ 𝒜 with h(0) = h′(0) − 1 = 0 satisfying Re h′(z) > 0 in 𝔻. For b ∈ 𝔻, we consider the sets For fixed z 0 ∈ 𝔻 \\ {0} and b ∈ 𝔻, 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.

The Properties of a New Subclass of Harmonic Univalent Mappings

We introduced a new subclass of univalent harmonic functions defined by the shear construction in the present paper. First, we showed that the convolutions of two special subclass harmonic mappings are convex in the horizontal direction. Secondly, we proved a necessary and sufficient condition for the above subclass of harmonic mappings to be convex in the horizontal direction. We also presented some basic examples of univalent harmonic functions explaining the behavior of the image domains.

On the linear combinations of harmonic univalent mappings

In this paper, we derive several sufficient conditions of the linear combinations of harmonic univalent mappings to be univalent and convex in the direction of the real axis. Furthermore, some illustrative examples and imagine domains of the linear combinations satisfying the desired conditions are enumerated.

Convolution Properties of Convex Harmonic Functions

In this paper, we examine the convolutions of convex harmonic functions with some other classes of univalent harmonic functions dened by certain coecient conditions and prove that such convolutions belong to some well known classes of univalent harmonic functions.

On Sakaguchi-Type Harmonic Univalent Functions

We consider the Sakaguchi functions which are starlike with respect to symmetrical points in the open unit disk U and extend it to class SH(;t ) concerning with Sakaguchi-type complex-valued harmonic univalent functions in U. A sucient

Harmonic maps between rotationally symmetric manifolds

AbstractWe prove the existence and uniqueness of harmonic maps between rotationally symmetric manifolds that are asymptotically hyperbolic.

A Subclass of Harmonic Univalent Functions with Varying Arguments Defined by Generalized Derivative Operator

Making use of the generalized derivative operator, we introduce a new class of complex valued harmonic functions which are orientation preserving and univalent in the open unit disc and are related to uniformly convex functions. We investigate the coefficient bounds, neighborhood, and extreme points for this generalized class of functions.

On the boundary behavior of univalent harmonic mappings

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A Convolution Approach on Partial Sums of Certain Harmonic Univalent Functions

The purpose of the present paper is to establish some new results giving the sharp bounds of the real parts of ratios of harmonic univalent functions to their sequences of partial sums by using convolution. Relevant connections of the results presented here with various known results are briefly indicated.

A New Subclass of Harmonic Univalent Functions Associated with Dziok-Srivastava Operator

The purpose of the present paper is to study a certain subclass of harmonic univalent functions associated with Dziok-Srivastava operator. We obtain coefficient conditions, distortion bounds, and extreme points for the above class of harmonic univalent functions belonging to this class and discuss a class preserving integral operator. We also show that class studied in this paper is closed under convolution and convex combination. The results obtained for the class reduced to the corresponding results for several known classes in the literature are briefly indicated.

A subclass of harmonic univalent functions with positive coefficients associated with fractional calculus operator

In the present paper, we are defining a subclass of harmonic univalent functions with positive coefficients in unit disc $U$ by using fractional calculus operator. Further we obtain distortion bounds, extreme points and radii of convexity for functions belonging to this class and discuss a class preserving integral operator. Here we also determine that the class studied in this paper is closed under convolution and convex combinations.

Subclass of Harmonic Univalent Functions Associated with Generalized Derivative Operator

Using a generalized derivative operator, we introduce and study a new subclass of harmonic univalent functions. In the present paper, we obtain numerous sharp results including coefficient conditions, extreme points, convolution properties and convex combinations for the above class of harmonic univalent functions. The results obtained for the class reduce to the corresponding results for various well-known classes in the literature.

Partial sums of certain harmonic univalent functions

Let ϕ(z) be a fixed harmonic functions of the form \(\varphi (z) = z + \sum\nolimits_{k = 2}^\infty {c_k z^k + } \overline {\sum\nolimits_{k = 1}^\infty {d_k z^k } } \) (dk ≥ ck ≥ c2 > 0; k ≥ 2) and SH(ck, dk, δ) be the subclass of harmonic univalent functions f(z) which satisfy the inequality \(\sum\nolimits_{k = 2}^\infty {c_k |a_k | + } \sum\nolimits_{k = 1}^\infty {d_k |b_k | \leqslant \delta } \). In this paper, we establish some interesting results on the ratio of harmonic univalent functions to its sequences of partial sums. Relevant connections of the results presented here with various known results are briefly indicated.