On Harmonic Functions Defined by Derivative Operator

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.

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.

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

Let be a harmonic univalent and orientation-preserving function on the unit disk, where h and g are analytic. We first give an example to show that there can be no upper-bound on the valency of h. Then we show that h is Bloch if, and only if, h is BMOA if, and only if, f is BMOA and we show that log h is Bloch if, and only if, log h is B MOA. Finally, we give sharp estimates for the Bloch-norm and BMOH2-norm of f when f maps the unit disk onto the unit disk.

ON SOME CLASSES OF BI-UNIVALENT FUNCTIONS

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.

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.

In this paper, we introduce a comprehensive family of harmonic univalent functions which contains various well-known classes of harmonic univalent functions as well as many new ones. Coefficient bounds, distortion bounds, extreme points, convolution conditions, and convex combination are determined for functions in this family. Consequently, many of our results are either extensions or new approaches to those corresponding previously known results.

This paper concerns a class ${\mathcal {P}_n}$ (defined below) of polynomials of degree less than or equal to $n$ having the properties: each polynomial which is univalent in the unit disk and of degree $n$ or less is in ${\mathcal {P}_n}$ and if $\{ {P_{{n_k}}}\} _{k = 1}^\infty$ is a sequence of polynomials such that ${P_{{n_k}}} \in {\mathcal {P}_{{n_k}}}$ and ${\lim _{k \to \infty }}{P_{{n_k}}} = f$ (uniformly on compact subsets of the unit disk) then $f$ is univalent. The approach is to study the extreme points in ${\mathcal {P}_n}$ ($P \in {\mathcal {P}_n}$ is extreme if $P$ is not a proper convex combination of two distinct elements of ${\mathcal {P}_n}$). Theorem 3 shows that if $P \in {\mathcal {P}_n}$ is extreme then $((n + 1)/n)P(z) - (1/n)zP’(z)$ is univalent and Theorem 6 gives a geometric condition on the image of the boundary of the disk under this mapping in order that $P$ be extreme. Theorem 10 states that the collection of polynomials univalent in the unit disk and having the property $P(z) = z + {a_2}{z^2} + \cdots + {a_n}{z^n},{a_n} = 1/n$, are dense in the class $S$ of normalized univalent functions. These polynomials have the very striking geometric property that the tangent line to the curve $P({e^{i\theta }})$, $0 \leqq \theta \leqq 2\pi$, turns at a constant rate (between cusps) as $\theta$ varies.

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.

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 study, we define new class of holomorphic functions associated with tangent function. Furthermore, we examine the differential subordination implementation results related to Janowski and tangent functions. Also, we investigate some extreme point theorem and partial sums results, necessary and sufficient conditions, convex combination, closure theorem, growth and distortion bounds, and radii of close-to-starlikeness and starlikeness for this newly defined functions class of holomorphic functions.

The class $S_{H}(p)$, $0 \\leq p \\lt 1$, of complex valued, meromorphic harmonic univalent sense-preserving functions in the unit disk $U \\backslash \\{p\\}$ is studied. The functions belong to $S_{H}(p)$ have the expansion $f(z) = \\frac{\\alpha}{z-p} + \\sum_{n=0}^{\\infty} c_{n} z^{n} + \\overline{\\sum_{n=1}^{\\infty} d_{n} z^{n}} + A\\log |z-p|$ and $\\lim_{z \\rightarrow p} f(z) = \\infty$. Some coefficient estimates, distortion and area theorems are obtained. Sufficient coefficient conditions for a class of meromorphic harmonic univalent sense-preserving functions that are starlike and convex are given.