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

In this paper, a class of complex-valued harmonic univalent functions f (z) in the open disk U is defined by fractional calculus operator. We obtained coefficient bounds, distortion inequalities, convex combination, extreme points and convolution conditions for this class.

In this paper, using Differential operator a certain subclass of harmonic univalent functions in the unit disc U = {z ∈ C : |z| < 1} is investigated. Some properties such as coefficient bounds, convex combination, extreme points and convolution conditions of this class are obtained. The newly introduced subclass of harmonic univalent functions is more general than the subclasses earlier found in the literature.

Let SH be the class of functions f = h+g that are harmonic univalent and sense-preserving in the open unit disk U = { z : |z| < 1} for which f (0) = f'(0)-1=0. In this paper, we introduce and study a subclass H( α, β) of the class SH and the subclass NH( α, β) with negative coefficients. We obtain basic results involving sufficient coefficient conditions for a function in the subclass H( α, β) and we show that these conditions are also necessary for negative coefficients, distortion bounds, extreme points, convolution and convex combinations. In this paper an attempt has also been made to discuss some results that uncover some of the connections of hypergeometric functions with a subclass of harmonic univalent functions.

We investigate some properties of a new subclass of harmonic univalent functions defined by multiplier transformation. We obtain coefficient conditions, extreme points, distortion bounds, convolution, and convex combination for the above subclass 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.

For , let denote the class of sense preserving harmonic mappings in the unit disk satisfying . The main aim of this paper is to study some basic properties such as coefficient bounds, growth estimates and convolution for functions in the class . We end the paper with an application, and construct harmonic univalent polynomials belonging to .

We define and investigate a special class of Salagean-type harmonic univalent functions in the open unit disk. We obtain coefficient conditions, extreme points, distortion bounds, convex combinations for the above class of harmonic univalent functions. 2000 Mathematics Subject Classification. Primary 30C45 ; Secondary 30C50

By applying Ruscheweyh derivative on the class ASH(λ, α, k, γ )o f harmonic univalent functions in the unit disk U , we obtain several interesting properties such as sharp coefficient relations, distortion bounds, extreme points, Hadamard product, and other results.

In this paper, we introduce a new class of harmonic univalent functions with respect to k-symmetric points by using a newly-defined q-analog of the derivative operator for complex harmonic functions. For this harmonic univalent function class, we derive a sufficient condition, a representation theorem, and a distortion theorem. We also apply a generalized q-Bernardi–Libera–Livingston integral operator to examine the closure properties and coefficient bounds. Furthermore, we highlight some known consequences of our main results. In the concluding part of the article, we have finally reiterated the well-demonstrated fact that the results presented in this article can easily be rewritten as the so-called (p,q)-variations by making some straightforward simplifications, and it will be an inconsequential exercise, simply because the additional parameter p is obviously unnecessary.

In this paper, we introduce certain subclasses of meromorphic harmonic univalent functions, which are defined by using generalized (<i>p, q</i>)-post quantum calculus operators as well as subordination relationship. Sufficient coefficient conditions, extreme points, distortion bounds and convolution properties for functions belonging to the subclasses are obtained.

We shall start with the analytic foundations and, building upon this, treat existence, regularity, and uniqueness of harmonic maps; in the course of that, already some geometric applications will occur. The A role of harmonic maps in Riemannian and Kahlerian geometry will then be discussed somewhat more systematically in the subsequent paragraphs. The last paragraph finally is devoted to a novel development of Teichmuller theory via harmonic maps.

We consider the class of all sense-preserving complex-valued harmonic mappings f = h + g ¯ f=h+\overline {g} defined on the unit disk D \mathbb {D} with the normalization h ( 0 ) = h ′ ( 0 ) − 1 = 0 h(0)=h’(0)-1=0 and g ( 0 ) = g ′ ( 0 ) = 0 g(0)=g’(0)=0 with the second complex dilatation ω : D → D \omega :\,\mathbb {D}\rightarrow \mathbb {D} , g ′ ( z ) = ω ( z ) h ′ ( z ) g’(z)=\omega (z)h’(z) . In this paper, the authors determine sufficient conditions on h h and ω \omega that would imply the univalence of harmonic mappings f = h + g ¯ f=h+\overline {g} on D \mathbb {D} .

In this paper, we obtain a new characterization for univalent harmonic mappings and obtain a structural formula for the associated function which defines the analytic -like functions in the unit disk. The new criterion stated in this article for the injectivity of harmonic mappings implies the well-known results of Kas’yanyuk [10] and Brickman [3] for analytic functions, but with a simpler proof than theirs. A number of consequences of the characterization, and examples are also presented. Further investigation provides a new method to construct univalent harmonic mappings with the help of an improved distortion theorem.

In the present paper, we study a subclass of univalent harmonic functions defined by convolution and integral convolution. We obtain the basic properties such as coefficient characterization and distortion theorem, extreme points and convolution condition.

Background The study of theory for analytic univalent and multivalent functions is an old subject in mathematics, particularly in complex analysis, that has captivated a great deal of scholars owing to the sheer sophistication of its geometrical features as well as its many research possibilities. The study of univalent functions is one of many significant elements of complex analysis for both single and multiple variables. Investigators have become keen on the conventional investigation of this topic since at least 1907. Numerous scholars in the area of complex analysis have emerged since then, including Euler, Gauss, Riemann, Cauchy, and other people. Geometric function theory combines geometry and analysis. Methods This study employs the differential subordination technique to derive multiple characteristics from the new linear operator M σ , μ n , ς Υ ( s ) . The concept of the differential subordination subclass of analytical univalent functions is analyzed. Results In this section, We studied some results on differential subordination and superordination using a specific class of univalent functions stated on a specific space of univalent functions stated on the open unit disc. Using properties of the operator, we discovered a number of properties of superordinations and subordinations related to the idea of the Hadamard product. We investigated several aspects of superordinations and subordinations using a new operator M σ , μ n , ς Υ ( s ) . Conclusions A new operator M σ , μ n , ς Υ ( s ) : Λ ⟶ Λ has been established in this paper connected to the Dziok-Srivastava operator T σ n and the Hadamard product corresponding to the Komatu integral operator Ω μ ς . The difference operator M σ , μ n , ς ϒ ( s ) can have specific properties derived by applying the differential subordination technique. And the objective of this paper is to make use of the connection ( β 1 μ + 1 ) M σ , μ n + 1 , ς ϒ ( s ) = w ( M σ , μ n , ς ϒ ( s ) ) ′ + β 1 μ ( M σ , μ n , ς ϒ ( s ) ) .