Sense-preserving Harmonic Mappings Research Articles

Stable classes of harmonic mappings

Let H0 denote the set of all sense-preserving harmonic mappings f=h+g‾ in the unit disk D, normalized with h(0)=g(0)=g′(0)=0 and h′(0)=1. In this paper, we mainly investigate some properties of certain subclasses of H0, including inclusion relations and stability analysis by precise examples, coefficient bounds, growth, covering and distortion theorems. As applications, we build some Bohr inequalities for these subclasses by means of subordination. Among these subclasses, six classes consist of functions f=h+g‾∈H0 such that h+ϵg is univalent (or convex) in D for each |ϵ|=1 (or for some |ϵ|=1, or for some |ϵ|≤1). Simple analysis shows that if the function f=h+g‾ belongs to a given class from these six classes, then the functions h+ϵg‾ belong to corresponding class for all |ϵ|=1. We call these classes as stable classes.

Radius of Convexity for Analytic Part of Sense-Preserving Harmonic Mappings

Radius of Convexity for Analytic Part of Sense-Preserving Harmonic Mappings

Quasiconformal extension of integral transforms of analytic and harmonic mappings

In this article, we prove a criterion for quasiconformal extension of the integral transforms of analytic functions defined in the unit disk of the complex plane. Thereafter, we prove a sufficient condition for quasiconformal extensibility of sense-preserving harmonic mappings defined in the unit disk and applying this result, we obtain criteria for the quasiconformal extension of various types of integral transforms that are defined for sense-preserving harmonic mappings. We also study the stability of the quasiconformal extension of sense-preserving harmonic mappings.

Injectivity of harmonic mappings with a specified injective holomorphic part

Let \(F=H+\overline{G}\) be a locally injective and sense-preserving harmonic mapping of the unit disk \(\mathbb{D}\) in the complex plane \(\mathbb{C}\), where \(H\) and \(G\) are holomorphic in \(\mathbb{D}\) and \(G(0)=0\). The aim of this paper is studying interplay between properties of \(F_\varepsilon:=H+\varepsilon\overline G\), \(\varepsilon\in\mathbb{C}\), and its holomorphic part \(H\). In particular, several results dealing with the injectivity of \(F_\varepsilon\) are obtained.

The Bohr inequality for certain harmonic mappings

Let HC(ϕ) and HCc(ϕ) denote the classes of sense-preserving harmonic mappings f=h+g¯ in D with the dilation g′(z)=αzh′(z) for |α|<1 such that h is Ma–Minda type convex function and Ma–Minda type convex function with respect to the conjugate points respectively. Here, the function ϕ:D→ℂ, called Ma–Minda function, is analytic and univalent in D such that ϕ(D) has positive real part, symmetric with respect to the real axis, starlike with respect to ϕ(0)=1 and ϕ′(0)>0. The classes HC(ϕ) and HCc(ϕ) are derived from the work of Sun et al. (2016) on the class M(α,β). We study the growth theorems for the functions in these classes. For general ϕ, the sharp coefficient bounds of the functions in the classes HC(ϕ) and HCc(ϕ) are not yet known. For the function f=h+g¯ of the form h(z)=z+∑n=2∞anzn and g(z)=∑n=2∞bnzn in D, using the Bohr phenomenon for subordination classes, we find the radius Rf<1 such that Bohr inequality |z|+∑n=2∞(|an|+|bn|)|z|n≤d(f(0),∂f(D))holds for |z|=r≤Rf for functions in the classes HC(ϕ) and HCc(ϕ) . As a consequence, we obtain several interesting corollaries on Bohr inequality for the aforesaid classes. In addition, we discuss the area theorem for functions in the class HC(ϕ) and obtain an improved version of Bohr inequality for functions in HC(ϕ).

Lower and upper order of harmonic mappings

In this paper, we define both the upper and lower order of a sense-preserving harmonic mapping in D. We generalize to the harmonic case some known results about holomorphic functions with positive lower order and we show some consequences of a function having finite upper order. In addition, we improve a related result in the case when there is equality in a known distortion theorem for harmonic mappings with finite upper order. Some examples are provided to illustrate the developed theory.

Criteria for univalency and quasiconformal extension for harmonic mappings

In this paper, we study the univalency and quasiconformal extension of sense-preserving harmonic mappings $f$ in the unit disk. For $f$, we introduce a quantity similar to Ahlfors's criteria and obtain a criterion of univalency and quasiconformal extension of $f$, which can be regarded as generalizations of the results obtained by Ahlfors [Sufficient conditions for quasiconformal extension, Ann. of Math. Stud. 79 (1974), 23-29], Hernández and Martín [Quasiconformal extensions of harmonic mappings in the plane, Ann. Acad. Sci. Fenn. Math. 38 (2013), 617-630], and Chen and Que [Quasiconformal extension of harmonic mappings with a complex parameter, J. Aust. Math. Soc. 102 (2017), 307-315]. By Schwarzian derivatives of harmonic mappings, we also obtain a criterion for univalency and quasiconformal extension for harmonic Techmüller mappings.

Boundary Schwarz Lemma for Harmonic Mappings Having Zero of Order p

Suppose $w$ is a sense-preserving harmonic mapping of the unit disk $\mathbb{D}$ such that $w(\mathbb{D})\subseteq\mathbb{D}$ and $w$ has a zero of order $p\geq1$ at $z=0$. In this paper, we first improve the Schwarz lemma for $w$, and then, we establish its boundary Schwarz lemma. Moreover, by using the automorphism of $\mathbb{D}$, we further generalize this result.

Quasiconformal Extensions of Harmonic Mappings

We derive a very general condition for a sense-preserving harmonic mapping with dilatation a square to be injective in the unit disk $${{\mathbb {D}}}$$ and to admit a quasiconformal extension to the extended complex plane. The analysis depends on geometric properties of an extension of the Weierstrass–Enneper lift to the extended plane that glues the parametrized minimal surface to a complementary topological hemisphere. The resulting topological sphere renders an entire graph over the complex plane provided additional restrictions on the dilatation are satisfied. The projection results in the desired extension. Several corollaries are drawn from the general criterion.

Bohr’s phenomenon for the classes of Quasi-subordination and K-quasiregular harmonic mappings

In this paper, we investigate the Bohr radius for $K$-quasiregular sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ such that the translated analytic part $h(z)-h(0)$ is quasi-subordinate to some analytic function. The main aim of this article is to extend and to establish sharp versions of four recent theorems by Liu and Ponnusamy \cite{LP2019} and, in particular, we settle affirmatively the two conjectures proposed by them. Furthermore, we establish two refined versions of Bohr's inequalities and determine the Bohr radius for the derivatives of analytic functions associated with quasi-subordination.

On convex combinations of harmonic mappings

Let psi_{mu,nu}(z)=(1-2cosnu e^{imu}z+e^{2imu}z^{2})^{-1}, mu,nuin[0,2pi) and p be an analytic mapping with operatorname{Re} p>0 on the open unit disk. We consider the sense-preserving planar harmonic mappings f=h+overline{g}, which are shears of the mapping int_{0}^{z} psi_{mu,nu}(xi) p(xi),{d}xi in the direction μ. These mappings include the harmonic right half-plan mappings, vertical strip mappings, and their rotations. For various choices of dilatations g'/h' of f, sufficient conditions are found for the convex combinations of these mappings to be univalent and convex in the direction μ.

Harmonic mappings with analytic part convex in one direction

In this paper, we study a family of sense-preserving harmonic mappings whose analytic part is convex in one direction. We first establish the bounds on the pre-Schwarzian norm. Next, we obtain radius of fully starlike and radius of fully convex for this family of harmonic mappings.

Bohr Phenomenon for Locally Univalent Functions and Logarithmic Power Series

In this article we prove Bohr inequalities for sense-preserving $K$-quasiconformal harmonic mappings defined in $\mathbb{D}$ and obtain the corresponding results for sense-preserving harmonic mappings by letting $K\to\infty$. One of the results includes the sharpened version of a theorem by Kayumov $\textit{et. al.}$ ($\textit{Math. Nachr.}$, 291 (2018), no. 11--12, 1757--1768). In addition Bohr inequalities have been established for uniformly locally univalent holomorphic functions, and for $\log(f(z)/z)$ where $f$ is univalent or inverse of a univalent function.

Bohr Radius for Subordination and K-quasiconformal Harmonic Mappings

The present article concerns the Bohr radius for K-quasiconformal sense-preserving harmonic mappings $$f=h+\overline{g}$$ in the unit disk $$\mathbb {D}$$ for which the analytic part h is subordinated to some analytic function $$\varphi $$ , and the purpose is to look into two cases: when $$\varphi $$ is convex, or a general univalent function in $$\mathbb {D}$$ . The results state that if $$h(z) =\sum _{n=0}^{\infty }a_n z^n$$ and $$g(z)=\sum _{n=1}^{\infty }b_n z^n$$ , then $$\begin{aligned} \sum _{n=1}^{\infty }(|a_n|+|b_n|)r^n\le {{\text {dist}}}(\varphi (0),\partial \varphi (\mathbb {D})) \quad \text{ for } r\le r^* \end{aligned}$$ and give estimates for the largest possible $$r^*$$ depending only on the geometric property of $$\varphi (\mathbb {D})$$ and the parameter K. Improved versions of the theorems are given for the case when $$b_1 = 0$$ and corollaries are drawn for the case when $$K\rightarrow \infty $$ .

A Valency Criterion for Harmonic Mappings

Let $$f=h+{\overline{g}}$$ be a sense-preserving harmonic mapping of the closed unit disk $$\overline{{\mathbb {D}}}$$ with a Blashke product dilatation $$B_m= g'/h'$$ of order m. The aim of this paper is to prove that if $$h'$$ has $$p-1$$ zeros, counting multiplicity, in $${\mathbb {D}}$$ and no zeros on $$\partial {\mathbb {D}},$$ and that $$\begin{aligned} {\text {Re}}\left\{ 1+\mathrm{{e}}^\mathrm{it}\frac{h''(\mathrm{{e}}^\mathrm{it})}{h'(\mathrm{{e}}^\mathrm{it})}\right\} >-\frac{1}{2}\sum _{k=1}^m\frac{1-|a_k|}{1+|a_k|}, \end{aligned}$$ where $$a_1,\ldots , a_m$$ are the zeros of $$B_m,$$ then f is $$(m+p-1)$$ -valent. The proof deploys a surface-theoretic technique based on an effective “pasting” procedure. This is an improvement of an earlier result of Bshouty et al. (Proc Am Math Soc 146:1113–1121, 2018) which asserts that if f is a sense-preserving harmonic mapping on $${\mathbb {D}},$$ with dilatation $$z^m$$ that satisfies the inequality $$\begin{aligned} {\text {Re}}\left\{ 1+ z\frac{h''(z)}{h'(z)}\right\} >-\frac{m}{2},\quad z\in {\mathbb {D}}, \end{aligned}$$ then f is $$(m+p)$$ -valent.

On a powered Bohr inequality

The object of this paper is to study the powered Bohr radius $\rho_p$, $p \in (1,2)$, of analytic functions $f(z)=\sum_{k=0}^{\infty} a_kz^k$ and such that $|f(z)|<1$ defined on the unit disk $|z|<1$. More precisely, if $M_p^f (r)=\sum_{k=0}^\infty |a_k|^p r^k$, then we show that $M_p^f (r)\leq 1$ for $r \leq r_p$ where $r_\rho$ is the powered Bohr radius for conformal automorphisms of the unit disk. This answers the open problem posed by Djakov and Ramanujan in 2000. A couple of other consequences of our approach is also stated, including an asymptotically sharp form of one of the results of Djakov and Ramanujan. In addition, we consider a similar problem for sense-preserving harmonic mappings in $|z|<1$. Finally, we conclude by stating the Bohr radius for the class of Bieberbach-Eilenberg functions.

Harmonic pre-Schwarzian and its applications

The primary aim of this article is to extend the study on the pre-Schwarzian derivatives from the case of univalent analytic functions to that of univalent harmonic mappings defined on certain domains. This is done in two different ways. One of the ways is to connect with a conjecture on the univalent harmonic mappings. Also, we improve certain known results on the majorization of the Jacobian of functions in the affine and linear invariant family of sense-preserving harmonic mappings. This is achieved as an application of a corresponding distortion theorem in terms of the harmonic pre-Schwarzian derivative.

Uniformly locally univalent harmonic mappings associated with the pre-Schwarzian norm

In this paper, we consider the class of uniformly locally univalent harmonic mappings in the unit disk and build a relationship between its pre-Schwarzian norm and uniformly hyperbolic radius. Also, we establish eight ways of characterizing uniformly locally univalent sense-preserving harmonic mappings. We also present some sharp distortions and growth estimates and investigate their connections with Hardy spaces. Finally, we study subordination principles of norm estimates.

Generalized harmonic Koebe functions

We present a family of sense-preserving harmonic mappings in the unit disk related to the classical generalized (analytic) Koebe functions. We prove that these are precisely the mappings that maximize simultaneously the real part of every Taylor coefficient as well as the growth and distortion of functions in affine and linear invariant families of complex-valued harmonic functions.

Harmonic K-quasiconformal Mappings from Unit Disk onto Half Planes

We consider the analytic properties for sense-preserving univalent harmonic mappings on the unit disk D onto half planes \(\Omega =\{w: \text {Re} w> a, -\infty 0, f_{\bar{z}}(0)\in R\). By deriving an analytic representing formula for these univalent harmonic mappings, we obtain one necessary and sufficient condition for these sense-preserving univalent harmonic mappings to be harmonic K-quasiconformal mappings, we also obtain sharp coefficient estimates for these harmonic K-quasiconformal mappings. As an application, distortion theorems on length and area for their images are also obtained. Our results improve and generalize the one made by M. Ozturk.