Univalent Harmonic Mappings Research Articles

On Linear Combinations Of Harmonic Mappings Convex In The Horizontal Direction

The process of creating univalent harmonic mappings which are not analytic is not simple or straightforward. One efficient method for constructing desired univalent harmonic maps is by taking the linear combination of two suitable harmonic maps. In this study, we take into account two harmonic, univalent, and convex in the horizontal direction mappings, which are horizontal shears of $\Psi_{m}(z)=\frac{1}{2i\sin \gamma_{m}}\log \left( \frac{ 1+ze^{i\gamma_{m}}}{% 1+ze^{-^{i\gamma_{m}}}}\right),$ and have dilatations $\omega _{1}(z)=z,$ $\omega _{2}(z)=\frac{z+b}{1+bz},$ $b\in (-1,1).$ We obtain sufficient conditions for the linear combination of these two harmonic mappings to be univalent and convex in the horizontal direction. In addition, we provide an example to illustrate the result graphically with the help of Maple.

Univalence of horizontal shear of Cesàro type transforms

This manuscript investigates the classical problem of determining conditions on the parameters α,β∈C for which the integral transformCαβ[φ](z):=∫0z(φ(ζ)ζ(1−ζ)β)αdζ is also univalent in the unit disk, where φ is a normalized univalent function. Additionally, whenever φ belongs to some subclasses of the class of univalent functions, the univalence features of the harmonic mappings corresponding to Cαβ[φ] and its rotations are derived. As applications to our primary findings, a few non-trivial univalent harmonic mappings are also provided. The primary tools employed in this manuscript are Becker's univalence criteria and the shear construction developed by Clunie and Sheil-Small.

A new subclass of harmonic univalent functions associated with q-calculus

The purpose of the present paper is to introduce a new subclass of harmonic univalent functions by applying q-calculus. Coefficient inequalities, extreme points, distortion bounds, covering results, convolution condition and convex combination are determined for this class. Finally, we discuss a class preserving integral operator for this class.

Integral mean estimates for univalent and locally univalent harmonic mappings

Abstract We verify a long-standing conjecture on the membership of univalent harmonic mappings in the Hardy space, whenever the functions have a “nice” analytic part. We also produce a coefficient estimate for these functions, which is in a sense best possible. The problem is then explored in a new direction, without the additional hypothesis. Interestingly, our ideas extend to certain classes of locally univalent harmonic mappings. Finally, we prove a Baernstein-type extremal result for the function $\log (h'+cg')$ , when $f=h+\overline {g}$ is a close-to-convex harmonic function, and c is a constant. This leads to a sharp coefficient inequality for these functions.

A Specific Category of Harmonic Functions Characterized By A Generalized Komatu Operator in Conjunction With The (R-K) Integral Operator and Applications to Neutrosophic Complex Field

In our work, we introduced a distinct subclass of univalent harmonic functions referred to as a subclass of chiral functions. These functions are defined by combining the generalized Komatu operator with the integral operator (R − K), which has positive coefficients within the unit disc A. Also, we generalize the same subclass into neutrosophic complex numbers. Throughout our investigation, we establish several properties associated with these functions, including coefficient estimates, the convex formula, the integral operator, and the Hadamard product. On the other hand, we present the Neutrosophic convex formula and the neutrosophic integral operator.

On Janowski Type Harmonic Functions Associated with the Wright Hypergeometric Functions

In our present study we consider Janowski type harmonic functions class introduced and studied by Dziok, whose members are given by $h(z) = z + \sum_{n=2}^{\infty} h_n z^n$ and $g(z) = \sum_{n=1}^{\infty} g_n z^n$, such that $\mathcal{ST}_{H}(F,G)=\big\{ f = h + \bar{g} \in {H}:\frac{\mathfrak{D}_H f(z)}{f(z)}\prec\frac{1+Fz}{1+G z};\, (-G \leq F < G \leq 1, \text{ with } g_1=0)\big\},$ where $\mathfrak{D}_H f(z) = zh'(z)-\overline{zg'(z)}\,$ and $z\in \mathbb{U}=\{z:z\in \mathbb{C} \text{ and }|z| < 1 \}.$ We investigate an~association between these subclasses of harmonic univalent functions by applying certain convolution operator concerning Wright's generalized hypergeometric functions and several special cases are given as a corollary. Moreover we pointed out certain connections between Janowski-type harmonic functions class involving the generalized Mittag–Leffler functions. Relevant connections of the results presented herewith various well-known results are briefly indicated.

On Bohr's inequality for special subclasses of stable starlike harmonic mappings

Abstract The focus of this article is to explore the Bohr inequality for a specific subset of harmonic starlike mappings introduced by Ghosh and Vasudevarao (Some basic properties of certain subclass of harmonic univalent functions, Complex Var. Elliptic Equ. 63 (2018), no. 12, 1687–1703.). This set is denoted as ℬ H 0 ( M ) ≔ { f = h + g ¯ ∈ ℋ 0 : ∣ z h ″ ( z ) ∣ ≤ M − ∣ z g ″ ( z ) ∣ } {{\mathcal{ {\mathcal B} }}}_{H}^{0}\left(M):= \{f=h+\overline{g}\in {{\mathcal{ {\mathcal H} }}}_{0}:| z{h}^{^{\prime\prime} }\left(z)| \le M-| z{g}^{^{\prime\prime} }\left(z)| \} for z ∈ D z\in {\mathbb{D}} , where 0 < M ≤ 1 0\lt M\le 1 . It is worth mentioning that the functions belonging to the class ℬ H 0 ( M ) {{\mathcal{ {\mathcal B} }}}_{H}^{0}\left(M) are recognized for their stability as starlike harmonic mappings. With this in mind, this research has a twofold goal: first, to determine the optimal Bohr radius for this specific subclass of harmonic mappings, and second, to extend the Bohr-Rogosinski phenomenon to the same subclass.

Estimation of the Bounds of Some Classes of Harmonic Functions with Symmetric Conjugate Points

In this paper, we introduce some classes of univalent harmonic functions with respect to the symmetric conjugate points by means of subordination, the analytic parts of which are reciprocal starlike (or convex) functions. Further, by combining with the graph of the function, we discuss the bound of the Bloch constant and the norm of the pre-Schwarzian derivative for the classes.

On Some Classes of Harmonic Functions Associated with the Janowski Function

In this paper, we introduce some classes of univalent harmonic functions with respect to the symmetric conjugate points by means of subordination, the analytic parts of which are reciprocal starlike (or convex) functions. Further, we discuss the geometric properties of the classes, such as the integral expression, coefficient estimation, distortion theorem, Jacobian estimation, growth estimates, and covering theorem.

Linear combination of convolution of harmonic univalent functions

Linear combination of convolution of harmonic univalent functions

Some Sub-Classes of Harmonic Univalent functions

Complex Analysis is branch of Geometric function theory. Geometric function theory concerned with interplay between the geometric properties of the image domain and analytic properties of the mapping functions. Some properties of analytic functions are exclusive and do not extend to more general harmonic mappings. In this paper we study the some subclasses of univalent harmonic functions like Coefficient Bounds, Distortion results and Convolution of Two functions.

Application of Integral Operator Generated by Touchard Polynomials to Certain Subclasses of Harmonic Functions

Let SH denote the class of functions f = h + g which are harmonic univalent and sense-preserving in the unite disk U = {z : |z| < 1} where h(z) = z +P∞ k=2 akzk, g(z) =∞Pk=1 bkzk (|b1| < 1). In this paper we establish connections between various subclasses of harmonic univalent functions by applying certain integral operator involving the Touchard Polynomials.

$q$-analogue of a class of harmonic functions

The purpose of the present paper is to introduce a new subclass of harmonic univalent functions associated with a $q$-Ruscheweyh derivative operator. A necessary and sufficient convolution condition for the functions to be in this class is obtained. Using this necessary and sufficient coefficient condition, results based on the extreme points, convexity and compactness for this class are also obtained.

On Harmonic Univalent Functions Defined by Dziok-Srivastava Operator

The purpose of this work is to present a class of harmonic univalent functions defined by the Dziok-Srivastava operator. Some geometric properties like coefficients conditions, distortion theorem, convolution (Hadamard product), convex combination and extreme points are investigated.
 2000 Mathematics Subject Classification: 30C45, 30C50

On the Bohr’s Inequality for Stable Mappings

We consider the class of stable harmonic mappings $$f=h+\overline{g}$$ introduced by Martin, Hernandez and the class of stable logharmonic mappings $$f=zh\overline{g}$$ introduced by AbdulHadi, El-Hajj. We determine Bohr’s radius for the classes of stable univalent harmonic mappings, stable convex harmonic mappings and stable univalent logharmonic mappings. We also consider improved and refined versions of Bohr’s inequality and discuss the Bohr–Rogosinski’s radius for this family of mappings.

A Study of Spiral-Like Harmonic Functions Associated with Quantum Calculus

This article introduces new subclasses of harmonic univalent functions associated with q -difference operator. Modified q -multiplier transformation is defined, and certain geometric properties such as the sufficient condition, distortion result, extreme points, and invariance of convex combination of the elements of the subclasses are discussed by employing the newly defined q -operator. Also, various well-known results already proved in the literature are pointed out.

A New Subclass of Harmonic Univalent Functions Defined by a Linear Operator

In this paper, a new subclass of complex-valued harmonic univalent functions f(z)=h(z)+g(z)¯¯¯¯¯¯¯¯¯ in the open disk U={z:z∈C and |z|<1} defined by using a linear operator. We investigate coefficient condition, extreme points, distortion and convex combination for this subclass.

Certain subclass of harmonic univalent functions defined by q-differential operator

In this paper, we define certain subclass of harmonic univalent function in the unit disc \(U = \left\{ {z \in C:\left| z \right| < 1} \right\}\) by using \(q\)-differential operator. Also we obtain coefficient inequalities, growth and distortion theorems for this subclass.

Convolution and Coefficient Estimates for (p,q)-Convex Harmonic Functions Associated with Subordination

We preface and examine classes of (p, q)-convex harmonic locally univalent functions associated with subordination. We acquired a coefficient characterization of (p, q)-convex harmonic univalent functions. We give necessary and sufficient convolution terms for the functions we will introduce.

Unconstrained Optimization of Univalent Harmonic Functions

The new generalized operator , is a conjunction between Unconstrained optimization and Univalent Harmonic Functions.We derived some properties by this conjunction like, coefficient inequality, convex set, apply Bernardi operator and determine the extreme points such that In particular, the extreme points of  (  are { } and { }.