Comprehensive family of harmonic univalent functions

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.

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.

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.

In this article, a new class of harmonic univalent functions, defined by the differential operator, is introduced. Some geometric properties, like, coefficient estimates, extreme points, convex combination and convolution (Hadamard product) are obtained.

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

In this paper, the computation of closed-form convex combinations is considered. In many control tasks, convex combinations play a crucial role, thus requiring an efficient computation. This is the case for online control of fast dynamical systems, in which the control algorithms rely on convex combinations, for example robust control of linear parameter-varying systems. On the other hand, for formal verification, it is necessary that the closed-loop behavior can be expressed in closed-form, which is not possible if the convex combination is expressed as constraints. In this paper, we provide closed-form expressions for any kind of polytope with finitely many extreme points. For special types of polytopes, such as simplices and parallelotopes, we provide especially efficient, analytical closed-form expressions. Numerical experiments show that the closed-form expressions are significantly faster in all randomly-generated cases and thus enable shorter sampling intervals of control schemes involving convex combinations.

Let denote the class of functions that are harmonic univalent and sense-preserv- ing in the unit disk , where . In this paper, we introduce the class of functions which are harmonic in . A sufficient coefficient of this class is determined. It is shown that this coefficient bound is also necessary for the class if , where and . Coefficient conditions, such as distortion bounds, convolution conditions, convex combination, extreme points, and neighborhood for the class , are obtained.

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, 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.

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.

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 a new subclass of harmonic functions that significantly improves our understanding of these functions in geometric function theory. We provide a comprehensive analysis of this subclass by deriving several important properties, including coefficient bounds and decay bounds, which are necessary to evaluate the behavior and limitations of functions in this class. Additionally, we establish sufficient coefficient conditions for harmonic functions to belong to this class. Moreover, we rigorously show that this subclass is closed under both convex combinations and convolutions, meaning that any convex combination or convolution of functions in this class will also belong to the class. These results provide valuable insights into the stability and applicability of the subclass and provide a solid framework for further theoretical explorations and practical applications in complex analysis.

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.

The aim of this paper is twofold. First, we investigate the properties of the composition of harmonic mappings with harmonic mappings, and the composition of biharmonic mappings with harmonic mappings. Second, we consider the Goodman-Saff conjecture for biharmonic mappings in the unit disk. In fact, we show that the answer to the Goodman-Saff conjecture is positive for a special class of univalently biharmonic mappings which contains the set of all harmonic univalent mappings.

We introduce new classes of meromorphic harmonic univalent functions. Using the duality principle, we obtain the duals of such classes of functions leading to coefficient bounds, extreme points and some applications for these functions.