Co-analytic Part Research Articles

Completely monotone sequences and harmonic mappings

In the present paper, we will study geometric properties of harmonic mappings whose analytic and co-analytic parts are (shifted) generated functions of completely monotone sequences.

Rosette Harmonic Mappings

A harmonic mapping is a univalent harmonic function of one complex variable. We define a family of harmonic mappings on the unit disk whose images are rotationally symmetric “rosettes” with n cusps or n nodes, \(n\ge 3\). These mappings are analogous to the n-cusped hypocycloid, but are modified by Gauss hypergeometric factors, both in the analytic and co-analytic parts. Relative rotations by an angle \(\beta \) of the analytic and anti-analytic parts lead to graphs that have cyclic, and in some cases dihedral symmetry of order n. While the graphs for different \(\beta \) can be dissimilar, the cusps are aligned along axes that are independent of \(\beta \). For certain isolated values of \(\beta \), the boundary function is continuous with arcs of constancy, and has nodes of interior angle \(\pi /2-\pi /n\) instead of cusps.

Geometric properties of harmonic functions associated with the symmetric conjecture points and exponential function

In this paper, some classes of univalent harmonic functions are introduced by subordination, where the analytic parts of which are exponential starlike (or convex) functions with respect to the symmetric conjecture points. According to the relationships of the analytic part and the co-analytic part, the geometric properties, such as coefficient estimates, distortion theorems, integral expressions, estimates and growth conditions and covering theorem, of the classes are obtained.

Some Classes of Harmonic Mapping with a Symmetric Conjecture Point Defined by Subordination

In the paper, we introduce some subclasses of harmonic mapping, the analytic part of which is related to general starlike (or convex) functions with a symmetric conjecture point defined by subordination. Using the conditions satisfied by the analytic part, we obtain the integral expressions, the coefficient estimates, distortion estimates and the growth estimates of the co-analytic part g, and Jacobian estimates, the growth estimates and covering theorem of the harmonic function f. Through the above research, the geometric properties of the classes are obtained. In particular, we draw figures of extremum functions to better reflect the geometric properties of the classes. For the first time, we introduce and obtain the properties of harmonic univalent functions with respect to symmetric conjugate points. The conclusion of this paper extends the original research.

Directional Convexity of Convolutions of Harmonic Functions

Harmonic functions can be constructed using two analytic functions acting as their analytic and coanalytic parts but the prediction of the behavior of convolution of harmonic functions, unlike the convolution of analytic functions, proved to be challenging. In this paper we use the shear construction of harmonic mappings and introduce dilatation conditions that guarantee the convolution of two harmonic functions to be harmonic and convex in the direction of imaginary axis.

Norm of the pre-Schwarzian derivative, Bloch's constant and coefficient bounds in some classes of harmonic mappings

The aim of the paper is a study of a family of sense-preserving complex valued harmonic functions f that are normalized in the open unit disk, and such that its analytic part is convex in one direction. We prove the univalence and establish estimates on pre-Schwarzian derivative and the Bloch constant for co-analytic part of harmonic mapping. The bounds of the coefficients of co-analytic part are also given.

Characterization of univalent harmonic mappings with integer or half-integer coefficients

Abstract Let 𝒮 H ${\mathcal{S}_{H}}$ denote the usual class of all normalized functions f = h + g ¯ ${f=h+\overline{g}}$ harmonic and sense-preserving univalent on the unit disk | z | < 1 ${|z|<1}$ . In this article we show that the set, consisting of those mappings f from 𝒮 H ${\mathcal{S}_{H}}$ for which all Taylor coefficients of the analytic and co-analytic parts of f are integers, consists of only nine functions. The second aim is to discuss the set 𝒮 H ${\mathcal{S}_{H}}$ of those functions which have half-integer coefficients. More precisely, we determine the set of univalent harmonic mappings with half-integer coefficients which are convex in real direction or convex in imaginary direction. This work generalizes the recent paper of Hiranuma and Sugawa. One of the examples generated in this way helps to disprove a conjecture of Bharanedhar and Ponnusamy.

Coefficient Estimates and Bloch’s Constant in Some Classes of Harmonic Mappings

Following Clunie and Sheil-Small, the class of normalized univalent harmonic mappings in the unit disk is denoted by $${\mathcal {S}}_{{\mathcal {H}}}$$ . The aim of the paper is to study the properties of a subclass of $${\mathcal {S}}_{{\mathcal {H}}}$$ , such that the analytic part is a convex function. We establish estimates of some functionals and bounds of the Bloch’s constant for co-analytic part.

Harmonic mappings for which co-analytic part is a close-to-convex function of order b

In the present paper we investigate a class of harmonic mappings for which the second dilatation is a close-to-convex function of complex order b, $b\in\mathbb{C} / \{ 0 \}$ (Lashin in Indian J. Pure Appl. Math. 34(7):1101-1108, 2003).

CLASSIFICATION OF UNIVALENT HARMONIC MAPPINGS ON THE UNIT DISK WITH HALF-INTEGER COEFFICIENTS

AbstractLet ${\mathcal{S}}$ denote the set of all univalent analytic functions $f$ of the form $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ on the unit disk $|z|<1$. In 1946, Friedman [‘Two theorems on Schlicht functions’, Duke Math. J.13 (1946), 171–177] found that the set ${\mathcal{S}}_{\mathbb{Z}}$ of those functions in ${\mathcal{S}}$ which have integer coefficients consists of only nine functions. In a recent paper, Hiranuma and Sugawa [‘Univalent functions with half-integer coefficients’, Comput. Methods Funct. Theory13(1) (2013), 133–151] proved that the similar set obtained for functions with half-integer coefficients consists of only 21 functions; that is, 12 more functions in addition to these nine functions of Friedman from the set ${\mathcal{S}}_{\mathbb{Z}}$. In this paper, we determine the class of all normalized sense-preserving univalent harmonic mappings $f$ on the unit disk with half-integer coefficients for the analytic and co-analytic parts of $f$. It is surprising to see that there are only 27 functions out of which only six functions in this class are not conformal. This settles the recent conjecture of the authors. We also prove a general result, which leads to a new conjecture.

New Subclasses of Harmonic Starlike and Convex Functions

Abstract. The purpose of the present paper is to establish some interesting results in-volving coefficient conditions, extreme points, distortion bounds and covering theoremsfor the classes V H (β) and U H (β). Further, various inclusion relations are also obtainedfor these classes. We also discuss a class preserving integral operator and show that theseclasses are closed under convolution and convex combinations. 1. IntroductionA continuous complex-valued function f = u + iv is said to be harmonic ina simply connected domain D if both u and v are real harmonic in D. In anysimply connected domain we can write f = h + g , where h and g are analytic inD. We call h the analytic part and g the co-analytic part of f. A necessary andsufficient condition for f to be locally univalent and sense-preserving in D is that h 0 (z) ,z> g 0 ∈ D. See Clunie and Sheil-Small [3], for more basic results onharmonic functions one may refer to the following standard introductory text bookby Duren [7], see also Ahuja [1] and Ponnusamy and Rasila ([9], [10]).Let S

Harmonic close-to-convex functions and minimal surfaces

In this paper, we study the family of sense-preserving complex-valued harmonic functions that are normalized close-to-convex functions on the open unit disk with . We derive a sufficient condition for to belong to the class . We take the analytic part of to be or and for a suitable choice of co-analytic part of , the second complex dilatation turns out to be a square of an analytic function. Hence, is lifted to a minimal surface expressed by an isothermal parameter. Explicit representation for classes of minimal surfaces are given. Graphs generated by using Mathematica are used for illustration.

On Harmonic Close-To-Convex Functions

In this paper, we study the family of sense-preserving complex-valued harmonic functions f that are normalized and close-to-convex on the open unit disk D. First we investigate the conditions for which f is close-to-convex on D. As a consequence, we derive a sufficient condition for f to be in this family. Using the condition, we establish sufficient conditions for f to be close-to-convex, in terms of the coefficients of the analytic and the co-analytic parts of f. Finally, we determine conditions on a, b such that \(f(z)=zF(a,b;a+b;z)+\overline{\alpha z^{2}F(a,b;a+b;z)}\) is harmonic close-to-convex (and hence univalent) in D, where F(a, b; c; z) denotes the Gaussian hypergeometric function. A similar result, and a number of interesting corollaries and examples of harmonic close-to-convex functions, are also obtained.

Which subnormal Toeplitz operators are either normal or analytic?

We study subnormal Toeplitz operators on the vector-valued Hardy space of the unit circle, along with an appropriate reformulation of P.R. Halmosʼs Problem 5: Which subnormal block Toeplitz operators are either normal or analytic? We extend and prove Abrahamseʼs theorem to the case of matrix-valued symbols; that is, we show that every subnormal block Toeplitz operator with bounded type symbol (i.e., a quotient of two bounded analytic functions), whose analytic and co-analytic parts have the “left coprime factorization”, is normal or analytic. We also prove that the left coprime factorization condition is essential. Finally, we examine a well-known conjecture, of whether every subnormal Toeplitz operator with finite rank self-commutator is normal or analytic.

Hyponormality and subnormality of block Toeplitz operators

In this paper, we are concerned with hyponormality and subnormality of block Toeplitz operators acting on the vector-valued Hardy space HCn2 of the unit circle.First, we establish a tractable and explicit criterion on the hyponormality of block Toeplitz operators having bounded type symbols via the triangularization theorem for compressions of the shift operator.Second, we consider the gap between hyponormality and subnormality for block Toeplitz operators. This is closely related to Halmos’s Problem 5: Is every subnormal Toeplitz operator either normal or analytic? We show that if Φ is a matrix-valued rational function whose co-analytic part has a coprime factorization then every hyponormal Toeplitz operator TΦ whose square is also hyponormal must be either normal or analytic.Third, using the subnormal theory of block Toeplitz operators, we give an answer to the following “Toeplitz completion” problem: find the unspecified Toeplitz entries of the partial block Toeplitz matrix A≔[U∗??U∗] so that A becomes subnormal, where U is the unilateral shift on H2.

Application of the subordination principle to the multivalent harmonic mappings with shear construction method

The harmonic function in the open unit disc D = { z ∈ C | | z | < 1 } can be written as a sum of an analytic and an anti-analytic function, f = h ( z ) + g ( z ) ¯ , where h ( z ) and g ( z ) are analytic functions in D , and are called the analytic part and co-analytic part of f , respectively. One of the most important questions in the study of the classes of such functions is related to bounds on the modulus of functions (growth) or modulus of the derivative (distortion), because the growth theorem and distortion theorem give the compactness of the classes of these functions. In this paper we consider both of these questions with the shear construction method.

Partial Sums of Starlike Harmonic Univalent Functions

Abstract. Although, interesting properties on the partial sums of analytic univalentfunctions have been investigated extensively by several researchers, yet analogous resultson partial sums of harmonic univalent functions have not been so far explored. The mainpurpose of the present paper is to establish some new and interesting results on the ratioof starlike harmonic univalent function to its sequences of partial sums. 1. IntroductionA continuous complex-valued function f = u+ ivis said to be harmonic ina simply connected domain D if both uand v are real harmonic in D. In anysimply connected domain we can writef = h+ g, where hand gare analytic inD. We call hthe analytic part and gthe co-analytic part of f:A necessary andsucient condition forf to be locally univalent and sense-preserving in Dis thatjh 0 (z)j>jg(z)j;z2D:See Clunie and Sheil-Small [2].Denote by S H the class of functions f= h+gwhich are harmonic univalent andsense-preserving in the unit disk U= fz: jzj<1gfor which f(0) = f

Application of the Subordination Principle to the Harmonic Mappings Convex in One Direction with Shear Construction Method

Any harmonic function in the open unit disc can be written as a sum of an analytic and antianalytic functions , where and are analytic functions in and are called the analytic part and the coanalytic part of respectively. Many important questions in the study of the classes of functions are related to bounds on the modulus of functions (growth) or the modulus of the derivative (distortion). In this paper, we consider both of these questions.

Joint Hyponormality of Rational Toeplitz Pairs

We characterize hyponormal “rational” Toeplitz pairs which are pairs of Toeplitz operators whose symbols are rational functions in L∞. The main result of this article is as follows. If T = (Tϕ, Tψ) is a hyponormal rational Toeplitz pair then ϕ − βψ ∈H2 for some constant β; in other words, their co-analytic parts necessarily coincide up to a constant multiple. As a corollary we get a complete characterization of hyponormal rational Toeplitz pairs.

Multivalent Harmonic Uniformly Starlike Functions

Abstract. In this paper, we investigate a generalized family of complex-valued harmonicfunctions that are multivalent, sense-preserving, and are associated with k-uniformly har-monic functions in the unit disk. The results obtained here include a number of knownand new results as their special cases. 1. IntroductionA harmonic function fde ned in a simply connected complex domain DˆCcan be expressed by f(z) = h(z) + g(z);z2D. We call hthe analytic part and gthe co-analytic part of f. If the co-analytic part of fis zero, then freduces to theanalytic case. The mapping z!f(z) is sense-preserving and locally one-to-one inDif and only if the Jacobian of fis positive, that is, if and only ifJ f (z) = jh 0 (z)j 2 j g 0 (z)j 2 >0; z2D:Denote by Hthe family of functions f= h+gwhich are harmonic, sense-preservingand univalent in the open unit disk 4= fz: jzj<1gwith(1.1) h(z) = z+X 1n=2 a n z n ; g(z) =X 1n=1 b n z n ; jb 1 j<1:The class H was de ned and studied by Clunie and Sheil-Small [10]. Also, see