Dorff et al. [4], proved that the harmonic convolution of right half-plane mapping with dilatation -z and mapping f_β= h_β+ \bar{g}_β, where f_βis obtained by shearing of analytic vertical strip mapping, with dilatation e^{iθ}z^n; n = 1,2,θ\in R, is in S_H^0 and is convex in the direction of the real axis. In this paper, by using Cohn's rule, we generalize this result by considering dilatations (a-z)/(1-az), a\in (-1,1) and e^{iθ} z^n (n\in N;θ\in R) of right half-plane mapping and f_β, respectively.
Convolutions or Hadamard products of analytic functions is a well explored area of research and many nice results are available in literature. On the other hand, very little is known in general about the convolutions of univalent harmonic mappings. So, researchers started exploring properties of convolutions of some specific univalent harmonic mappings and while doing so, they have mostly used well known Cohn's rule or/and Schur-Cohn's algorithm, which involves computations that are very cumbersome. The main objective of this article is to present an alternative approach, which requires very l…
We use the Baernstein star-function to investigate several questions about the integral means of the convolution of two analytic functions in the unit disc. The theory of univalent functions plays a basic role in our work.
We consider the convolution of half-plane harmonic mappings with respective dilatations $(z+a)/(1+az)$ and $e^{iθ}z^{n}$, where $-1<a<1$ and $θ\in\mathbb{R},n\in\mathbb{N}$. We prove that such convolutions are locally univalent for $n=1$, which solves an open problem of Dorff et. al (see \cite[Problem~3.26]{Bshouty2010}). Moreover, we provide some numerical computations to illustrate that such convolutions are not univalent for $n\geq 2$.
Let $\ast$ and $\widetilde {\ast}$ denote the convolution of two analytic maps and that of an analytic map and a harmonic map respectively. Pokhrel [1] proved that if $f = h+\overline{g}$ is a harmonic map convex in the direction of $e^{iγ}$ and $ϕ$ is an analytic map in the class DCP, then $f\widetilde{\ast} ϕ= h\widetilde{\ast}ϕ+ \overline{g\widetilde{\ast}ϕ}$ is also convex in the direction of $e^{iγ}$, provided $f\widetilde{\ast}ϕ$ is locally univalent and sense-preserving. In the present paper we obtain a general condition under which $f\widetilde{\ast} ϕ$ is locally univalent and sense-p…
Let $E$ be the open unit disk $\{z\in \mathbb{C}: |z|<1\}$. Let $A$ be the class of analytic functions in $E$, which have the form $f(z)=z+a_2z^2+...$. We define operators $L_n^σ\colon A\to A$ using the convolution *. Using these operators, we define and study new classes of functions in the unit disk. Moreover, we obtain some basic properties of the new classes, namely inclusion, growth, covering, distortion, closure under certain integral transformation and coefficient inequalities.
In the present paper, we investigate connections between various subclasses of harmonic univalent functions by using a convolution operator involving the Pascal distribution series.
The first author proved that the harmonic convolution of a normalized right half-plane mapping with either another normalized right half-plane mapping or a normalized vertical strip mapping is convex in the direction of the real axis. provided that it is locally univalent. In this paper, we prove that in general the assumption of local univalency cannot be omitted. However, we are able to show that in some cases these harmonic convolutions are locally univalent. Using this we obtain interesting examples of univalent harmonic maps one of which is a map onto the plane with two parallel slits.
Dorff, proved in [2] that the convolution of two harmonic right-half plane mappings is convex in the direction of real axis provided that the convolution is locally univalent and sense preserving. Later, it was shown in [3] that the condition of locally univalent and sense preserving can be dropped in some special cases. In this paper, we generalize the main result from [3].
In the present paper, we derive several conditions of linear combinations and convolutions of harmonic mappings to be univalent and convex in one direction, one of them gives a partial answer to an open problem proposed by Dorff. The results presented here provide extensions and improvements of those given in some earlier works. Several examples of univalent harmonic mappings convex in one direction are also constructed to demonstrate the main results.
Let ${\mathcal S^0}(H_γ)$ denote the class of all univalent, harmonic, sense-preserving and normalized mappings $f$ of the unit disk $\ID$ onto the slanted half-plane $H_γ:=\{w:\,{\rm Re\,}(e^{iγ}w) >-1/2\}$ with an additional condition $f_{\bar{z}}(0)=0$. Functions in this class can be constructed by the shear construction due to Clunie and Sheil-Small which allows by examining their conformal counterpart. Unlike the conformal case, convolution of two univalent harmonic convex mappings in $\ID$ is not necessarily even univalent in $\ID$. In this paper, we fix $f_0\in{\mathcal S^0}(H_{0})$ and…
The hereditary property of convexity and starlikeness for conformal mappings does not generalize to univalent harmonic mappings. This failure leads us to the notion of fully starlike and convex mappings of order α, (0\leq α<1). A bound for the radius of fully starlikeness and fully convexity of order αis determined for certain families of univalent harmonic mappings. Convexity is not preserved under the convolution of univalent harmonic convex mappings, unlike in the analytic case. Given two univalent harmonic convex mappings f and g, the problem of finding the radius r_{0} such that f*g is a …
In this paper, we obtain coefficient criteria for a normalized harmonic function defined in the unit disk to be close-to-convex and fully starlike, respectively. Using these coefficient conditions, we present different classes of harmonic close-to-convex (resp. fully starlike) functions involving Gaussian hypergeometric functions. In addition, we present a convolution characterization for a class of univalent harmonic functions discussed recently by Mocanu, and later by Bshouty and Lyzzaik in 2010. Our approach provide examples of harmonic polynomials that are close-to-convex and starlike, res…
The convolution properties are discussed for the complex-valued harmonic functions in the unit disk $\mathbb{D}$ constructed from the harmonic shearing of the analytic function $ϕ(z):=\int_0^z (1/(1-2ξ\textit{e}^{\textit{i}μ}\cosν+ξ^2\textit{e}^{2\textit{i}μ}))\textit{d}ξ$, where $μ$ and $ν$ are real numbers. For any real number $α$ and harmonic function $f=h+\overline{g}$, define an analytic function $f_α:=h+\textit{e}^{-2\textit{i}α}g$. Let $μ_1$ and $μ_2$ $(μ_1+μ_2=μ)$ be real numbers, and $f=h+\overline{g}$ and $F=H+\overline{G}$ be locally-univalent and sense-preserving harmonic functions…
Let $\mathcal{H}$ denote the class of all complex-valued harmonic functions $f$ in the open unit disk normalized by $f(0)=0=f_{z}(0)-1=f_{\bar{z}}(0)$, and let $\mathcal{A}$ be the subclass of $\mathcal{H}$ consisting of normalized analytic functions. For $ϕ\in \mathcal{A}$, let $\mathcal{W}_{H}^{-}(ϕ):=\{f=h+\bar{g} \in \mathcal{H}:h-g=ϕ\}$ and $\mathcal{W}_{H}^{+}(ϕ):=\{f=h+\bar{g} \in \mathcal{H}:h+g=ϕ\}$ be subfamilies of $\mathcal{H}$. In this paper, we shall determine the conditions under which the harmonic convolution $f_1*f_2$ is univalent and convex in one direction if $f_1 \in \mathc…
This paper makes a modest contribution to the family of starlike univalent mappings in the open unit disk, by the introduction of some new subclasses of them via certain convolution operators. A new univalence condition is given with examples. Some basic characterizations of functions of the new subclasses are also mentioned.
In the article the authors consider the class ${\mathcal H}_0$ of sense-preserving harmonic functions $f=h+\overline{g}$ defined in the unit disk $|z|<1$ and normalized so that $h(0)=0=h'(0)-1$ and $g(0)=0=g'(0)$, where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes $\mathcal{P}_H^0(α)$ and $\mathcal{G}_H^0(β)$ of functions from ${\mathcal H}_0$ and show that if $f\in \mathcal{P}_H^0(α)$ and $F\in\mathcal{G}_H^0(β)$, then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions…
A subclass of complex-valued close-to-convex harmonic functions that are univalent and sense-preserving in the open unit disc is investigated. The coefficient estimates, growth results, area theorem, boundary behavior, convolution and convex combination properties for the above family of harmonic functions are obtained.
We show that the convolution of the harmonic function $f=h+\bar{g}$, where $h(z)+{e}^{-2{i}γ}g(z)=z/(1-{e}^{{i}γ}z)$ having analytic dilatation ${e}^{{i}θ} z^n (0\leqθ<2π)$, with the mapping $f_{a,α}=h_{a,α}+\overline{g}_{a,α}$, where $h_{a,α}(z)=(z/(1+a)-{e}^{{i}α}z^2/2)/(1-{e}^{{i}α}z)^2$, $g_{a,α}(z)=(a {e}^{2{i}α}z/(1+a)-{e}^{3{i}α}z^2/2)/(1-{e}^{{i}α}z)^2$ is convex in the direction $-(α+γ)$. We also show that the convolution of $f_{a,α}$ with the right half-plane mapping having dilatation $(a-z^2)/(1-az^2)$ is convex in the direction $-α$. Finally, we introduce a family of univalent harm…
In this article, we introduce a new family of sense preserving harmonic mappings f in the open unit disk and prove that functions in this family are close-to-convex. We give some basic properties such as coefficient bounds, growth estimates, convolution and determine the radius of convexity for the functions belonging to this family. In addition, we construct certain harmonic univalent polynomials belonging to this family.
For $α> -1$ and $β>0, $ let $\mathcal{B}_{\mathcal{H}}^0(α, β)$ denote the class of sense preserving harmonic mappings $f=h+\overline{g}$ in the open unit disk $\mathbb{D}$ satisfying $|zh''(z)+α(h'(z)-1)|\leq β-|zg''(z)+αg'(z)|.$ First, we establish that each function belonging to this class is close-to-convex in the open unit disk if $β\in (0, 1+α]$. Next, we obtain coefficient bounds, growth estimates and convolution properties. We end the paper with applications and will construct harmonic univalent polynomials belonging to this class.
