Functions In Half-plane Research Articles

Blaschke Products and Convolutions with a Slanted Generalized Half-Plane Harmonic Mapping

It is known that if the convolution of two suitably normalized planar harmonic mappings from certain families of mappings, such as those mapping into a half-plane or a strip, is locally univalent, the convolution is univalent and convex in one direction. After extending this to convolutions of mappings into a slanted half-plane with those into a slanted asymmetric strip, we prove properties for the dilatation of the convolution of a mapping from a family of slanted generalized right half-plane mappings with mappings into a slanted half-plane or a slanted asymmetric strip with a finite Blaschke product dilatation. The properties lay the foundation for a direct application of polynomial zero distribution techniques in the determination of local univalence of such convolutions. We conclude by producing a family of univalent convolutions convex in one direction between a slanted generalized right half-plane mapping and a mapping into a half-plane with a two-factor Blaschke product dilatation.

Combinations of Right Half-Plane Mappings and Vertical Strip Mappings Convex in the Vertical Direction

Combinations of Right Half-Plane Mappings and Vertical Strip Mappings Convex in the Vertical Direction

On univalent log-harmonic mappings

We consider the class of univalent log-harmonic mappings on the unit disk. Firstly, we present general idea of constructing log-harmonic Koebe mappings, log-harmonic right half-plane mappings and log-harmonic two-slits mappings and then we show precise ranges of these mappings. Moreover, coefficient estimates for univalent log-harmonic starlike mappings are obtained. Growth and distortion theorems for certain special subclasses of log-harmonic mappings are studied. Finally, we propose two conjectures, namely, log-harmonic coefficient and log-harmonic covering conjectures.

Rotations and convolutions of harmonic convex mappings

In this paper, we mainly consider the convolutions of slanted half-plane mappings and strip mappings of the unit disk D. If f1 is a slanted half-plane mapping and f2 is a slanted half-plane mapping or a strip mapping, then we prove that f1 * f2 is convex in some direction if f1 * f2 is locally univalent in D. We also obtain two sufficient conditions for f1 * f2 to be locally univalent in D. Our results extend many of the recent results in this direction. Moreover, we consider a class of harmonic mappings including slanted half-plane mappings and strip mappings, and as a consequence, we prove that the any convex combination of such locally univalent and sense-preserving mappings is also convex.

A proof of a conjecture on convolution of harmonic mappings and some related problems

UDC 517.5 Recently, Kumar et al. proposed a conjecture concerning the convolution of a generalized right half-plane mapping with a vertical strip mapping. They have verified the above conjecture for and . Also, it has been proved only for . In this paper, by using of a new method, we settle this conjecture in the affirmative for all and . Moreover, we will use this method to prove some results on convolution of harmonic mappings. This new method simplifies calculations and shortens the proof of results remarkably.

On convolutions of slanted half-plane mappings

The convolution of convex harmonic univalent functions in the unit disk, unlike analytic functions, may not be convex or even univalent. The main purpose of this work is to develop previous work involving the convolution of convex harmonic functions. Briefly, we obtain under which conditions the convolution of a right half-plane harmonic mapping having a dilatation and a slanted half-plane harmonic mapping with β having a dilatation and is univalent and convex in the direction . We also provide an example illustrating graphically with the help of Maple to illuminate the result.

Convolutions of Normalized Harmonic Mappings

Recent results on the convolution of two planar harmonic mappings is built upon the theory that when the convolution of functions from certain families of mappings, such as half-plane or strip mappings, is locally univalent, then the convolution will possess certain direction-convexity properties. Thus, much of the latest work on harmonic convolutions centers around establishing conditions on the dilatations of $$f_1, f_2: {\mathbb D}\rightarrow {\mathbb C}$$ from the families above so that $$f_1 * f_2$$ is locally univalent. Recently, it was noted that normalizations for these families were not treated properly when some dilatations considered did not fix zero. In this paper, we account for a variety of dilatations that do not fix zero by broadening the family from which $$f_1$$ and $$f_2$$ are chosen. Additionally, we show that when removing a hypothesis from one of our results it is possible to have a locally univalent convolution that fails to be univalent. This demonstrates that some of the previous work on convolutions cannot simply be modified by a re-normalization while affirming the necessity of the hypothesis.

NOTE ON THE CONVOLUTION OF HARMONIC MAPPINGS

Dorff et al. [‘Convolutions of harmonic convex mappings’, Complex Var. Elliptic Equ. 57(5) (2012), 489–503] formulated a question concerning the convolution of two right half-plane mappings, where the normalisation of the functions was considered incorrectly. In this paper, we reformulate the problem correctly and provide a solution to it in a more general form. We also obtain two new theorems which correct and improve related results.

Convolutions of Harmonic Right Half-Plane Mappings with Harmonic Strip Mappings

We prove that convolutions of harmonic right half-plane mappings with harmonic vertical strip mappings are univalent and convex in the horizontal direction. The proofs of these results make use the Gauss–Lucas Theorem. Our results show that two recent conjectures, the one by Kumar, Gupta, Singh and Dorff, and the one of Liu, Jiang and Li, are true. Moreover, examples of univalent harmonic mappings related to the above-mentioned results are presented, suggesting that the bounds given by our results may be sharp.

Generalized Characteristics for Meromorphic in the Half-plane Functions

In this paper we introduce generalized characteristics for meromorphic in the halfplane functions, and generalize the Levin’s formula and the first fundamental theorem for Tsuji’s characteristics.

A boundary property of some subclasses of functions of bounded type in the half-plane

Abstract The paper gives the construction of the half-plane analog of the part of the factorization theory of M.M. Djrbashian – V.S. Zakaryan, where Djrbashian’s generalized fractional integral was used to establish the descriptive representations and boundary properties of meromorphic in the unit disc functions of the classes N{ω} contained in the Nevanlinna class N of functions of bounded type. Some results of nearly the same type are obtained for several weighted classes of meromorphic in the upper half-plane functions with bounded Tsuji characteristics by application of the Laplace transform along with an Hadamard–Liouville type generalized integro-differential operator with an unbounded integration contour, which becomes the Liouville integro-differentiation in a particular case.

On Boundedness of Bergman Projection Operators in Banach Spaces of Holomorphic Functions in Half-Plane and Harmonic Functions in Half-Space

We present a simple proof of the boundedness of holomorphic and harmonic Bergman projection operators on a half-plane and a half-space respectively on the Orlicz space, the variable exponent Lebesgue space, and the variable exponent generalized Morrey space. The approach is based on an idea due to V. P. Zaharyuta and V. I. Yudovich (1962) to use Calderon–Zygmund operators for proving the boundedness of the Bergman projection in Lebesgue spaces on the unit disc. We also study the rate of growth of functions near the boundary in the spaces under consideration.

Convolutions of harmonic half-plane mappings with harmonic vertical strip mappings

In the present paper, we prove the convolutions of generalized harmonic right half-plane mappings with harmonic vertical strip mappings are univalent and convex in the horizontal direction. Moreover, some examples of harmonic univalent mappings convex in the horizontal direction are also constructed to illuminate the main results.

Univalency of Convolutions of Univalent Harmonic Right Half-Plane Mappings

We consider the convolution of half-plane harmonic mappings with respective dilatations \((z+a)/(1+az)\) and \(e^{i\theta }z^{n}\), where \(-1<a<1\) and \(\theta \in \mathbb {R},n\in \mathbb {N}\). We prove that such convolutions are locally univalent for \(n=1\), which solves an open problem of Dorff et al. (see J Anal 18:69–81 [3, Problem 3.26]). Moreover, we provide some numerical computations to illustrate that such convolutions are not univalent for \(n\ge 2\).

Convex Combinations of Planar Harmonic Mappings Realized Through Convolutions with Half-Strip Mappings

Recent investigations into what geometric properties are preserved under the convolution of two planar harmonic mappings on the open unit disk $${\mathbb {D}}$$ have typically involved half-plane and strip mappings. These results rely on having a convolution that is locally univalent and sense-preserving on $${\mathbb {D}}$$ , and thus, much focus has been on trying to satisfy this condition. We introduce a family of right half-strip harmonic mappings, $$\Psi _c : {\mathbb {D}}\rightarrow {\mathbb {C}}$$ , $$c>0$$ , and consider the convolution $$\Psi _c * f$$ for a harmonic mapping $$f = h +\overline{g}: {\mathbb {D}}\rightarrow {\mathbb {C}}$$ . We prove it is sufficient for $$h \pm g$$ to be starlike for $$\Psi _c *f$$ to be locally univalent and sense-preserving. Moreover, $$\Psi _c * f$$ decomposes into a convex combination of two harmonic mappings, one of which is f itself. This decomposition is key in addressing mapping properties of the convolution, and from it, we produce a family of convex octagonal harmonic mappings as well some other families of convex harmonic mappings. Additionally, motivated by the construction of $$\Psi _c$$ , we introduce a generalized harmonic Bernardi integral operator. We demonstrate convolution preserving properties and a weak subordination relationship for this extended operator.

Convolutions of harmonic right half-plane mappings

Abstract We first prove that the convolution of a normalized right half-plane mapping with another subclass of normalized right half-plane mappings with the dilatation − z ( a + z ) / ( 1 + a z ) $ - z(a + z)/(1 + az)$ is CHD (convex in the horizontal direction) provided a = 1 $a = 1$ or − 1 ≤ a ≤ 0 $ - 1 \le a \le 0$ . Secondly, we give a simply method to prove the convolution of two special subclasses of harmonic univalent mappings in the right half-plane is CHD which was proved by Kumar et al. [1, Theorem 2.2]. In addition, we derive the convolution of harmonic univalent mappings involving the generalized harmonic right half-plane mappings is CHD. Finally, we present two examples of harmonic mappings to illuminate our main results.

Convolution Properties of Some Harmonic Mappings in the Right Half-Plane

It is well known that harmonic convolution of two normalized right half-plane mappings is convex in the direction of the real axis, provided the convolution function is locally univalent and sense-preserving in \(E = \{z: |z|<1\}\). Further, it is also known that the condition of local univalence and sense-preserving in E on the convolution function can be dropped when one of the convoluting functions is the standard right half-plane mapping with dilatation \(\displaystyle -z\) and other is the right half-plane mapping with dilatation \(e^{i\theta } z^n,\,n=1,2,\)\(\theta \in \mathbb {R}.\) This result does not hold for \(n=3,4,5,\ldots .\) In this paper, we generalize this result by taking the dilatation of one of the right half-plane mappings as \(e^{i\theta } z^n\)\((\,n\in \mathbb {N},\theta \in \mathbb {R})\) and that of the other as \(\displaystyle {(a-z)}/{(1-az)},\)\(a\in (-1,1).\) We shall prove that our result holds true for all \(n\in \mathbb {N},\) provided the real constant a is restricted in the interval \(\left[ {(n-2)}/{(n+2)},1\right) \). The range of the real constant a is shown to be sharp.

Sections of stable harmonic convex functions

In this article, we consider the class SHC of normalized stable harmonic convex mappings f=h+g¯ in the unit disk and determine r such that every section of f is close-to-convex or convex in the disk ∣z∣<r. Also, we show that the convolution f∗f∈SHC whenever f∈SHC. In addition, we prove that the harmonic convolution f1∗f2 is stable harmonic close-to-convex whenever f1∈SHC and f2 is either a slanted half-plane mapping or an asymmetric vertical strip mapping or a mapping in the family PH0(α) which is contained in the family of harmonic close-to-convex functions in ∣z∣<1.

Convolutions of Harmonic Mappings Convex in One Direction

Let $$f_1$$ be a fixed univalent and sense-preserving harmonic mapping which is convex in the real direction but is not starlike in the unit disk $$|z|<1$$ . The convolution of $$f_1$$ with other harmonic mappings, eg. half-plane mappings, is not necessarily univalent in $$|z|<1$$ . In this paper, under suitable restriction on the dilatation of $$f$$ , we show that the convolutions of $$f_1$$ with certain slanted half-plane harmonic mappings $$f\in {\mathcal S^0}(H_{\gamma })$$ are necessarily convex in a direction. In addition, we consider a fixed harmonic mapping $$f_0$$ and $$f=h+\overline{g}\in {\mathcal S^0}(H_{\gamma })\cup {\mathcal S^0}(\Omega _\alpha )$$ with the dilatation $$\omega (z)=az^n$$ , where $${\mathcal S^0}(\Omega _\alpha )$$ denotes the class of asymmetric vertical strip mappings. We find the relationship between $$a$$ and $$n$$ such that $$f_0*f$$ is a sense-preserving univalent harmonic mapping and is convex in some direction. These results are a generalization of the corresponding recent result of Dorff et al.. The contents of this paper enhance the interest in univalent harmonic mappings, especially when much is not known on the harmonic convolution.

Univalency of convolutions of harmonic mappings

We consider the convolution or Hadamard product of planar harmonic mappings that are the vertical shears of the canonical half-plane mapping ? ( z ) = z / ( 1 - z ) with respective dilatations - xz and - yz , where | x | = | y | = 1 . We prove that any such convolution is univalent. Furthermore, in the case that x = y = - 1 , we show the resulting convolution is convex.