scholarly journals A Schwarz Lemma for Harmonic Mappings Between the Unit Balls in Real Euclidean Spaces

2018 ◽  
Vol 39 (6) ◽  
pp. 1065-1092
Author(s):  
Shaoyu Dai ◽  
Yifei Pan
Keyword(s):  
Schwarz Lemma ◽  
Harmonic Mappings ◽  
Euclidean Spaces

scholarly journals A Schwarz lemma for hyperbolic harmonic mappings in the unit ball

2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Jiaolong Chen ◽  
David Kalaj
Keyword(s):  
Unit Ball ◽  
Explicit Form ◽  
Smooth Function ◽  
Harmonic Mapping ◽  
Sharp Inequality ◽  
Schwarz Lemma ◽  
Harmonic Mappings ◽  
Manuscripta Math ◽  
Sharp Constant ◽  
Mapping Theory

Assume that $p\in [1,\infty ]$ and $u=P_{h}[\phi ]$, where $\phi \in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^n)$ and $u(0) = 0$. Then we obtain the sharp inequality $\lvert u(x) \rvert \le G_p(\lvert x \rvert )\lVert \phi \rVert_{L^{p}}$ for some smooth function $G_p$ vanishing at $0$. Moreover, we obtain an explicit form of the sharp constant $C_p$ in the inequality $\lVert Du(0)\rVert \le C_p\lVert \phi \rVert \le C_p\lVert \phi \rVert_{L^{p}}$. These two results generalize and extend some known results from the harmonic mapping theory (D. Kalaj, Complex Anal. Oper. Theory 12 (2018), 545–554, Theorem 2.1) and the hyperbolic harmonic theory (B. Burgeth, Manuscripta Math. 77 (1992), 283–291, Theorem 1).

scholarly journals Some Properties and Regions of Variability of Affine Harmonic Mappings and Affine Biharmonic Mappings

2009 ◽  
Vol 2009 ◽  
pp. 1-14
Author(s):  
Sh. Chen ◽  
S. Ponnusamy ◽  
X. Wang
Keyword(s):  
Analytic Functions ◽  
Schwarz Lemma ◽  
Harmonic Mappings

We first obtain the relations of local univalency, convexity, and linear connectedness between analytic functions and their corresponding affine harmonic mappings. In addition, the paper deals with the regions of variability of values of affine harmonic and biharmonic mappings. The regions (their boundaries) are determined explicitly and the proofs rely on Schwarz lemma or subordination.

Schwarz lemma in Euclidean spaces

2006 ◽  
Vol 51 (7) ◽  
pp. 653-659 ◽  
Author(s):  
Yan Yang ◽  
Tao Qian
Keyword(s):  
Schwarz Lemma ◽  
Euclidean Spaces

scholarly journals Quasihyperbolic quasi-isometry and Schwarz lemma of planar flat harmonic mappings

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5371-5383
Author(s):  
Qingtian Shi ◽  
Yi Qi
Keyword(s):  
Quasiconformal Mapping ◽  
Harmonic Mapping ◽  
Schwarz Lemma ◽  
Harmonic Mappings ◽  
Sufficient Condition

A sufficient condition of a flat harmonic quasiconformal mapping to be a quasihyperbolic quasiisometry on any subdomain of C is given in this paper, which generalizes the corresponding results of Euclidean and 1/|?|2 harmonic mappings. As an application, Schwarz lemma of flat harmonic mapping is also investigated. Besides, properties and constructions of flat harmonic mapping are obtained at the same time.

A Class of Harmonic Starlike Functions defined by Multiplier Transformation

2020 ◽  
pp. 455-469
Author(s):  
Deepali Khurana ◽  
Raj Kumar ◽  
Sibel Yalcin
Keyword(s):  
Convex Combination ◽  
Univalent Functions ◽  
Extreme Points ◽  
Starlike Functions ◽  
Harmonic Mappings ◽  
Distortion Bounds ◽  
Radius Of Convexity ◽  
Univalent Harmonic Mappings ◽  
Harmonic Starlike ◽  
Multiplier Transformation

We define two new subclasses, $HS(k, \lambda, b, \alpha)$ and \linebreak $\overline{HS}(k, \lambda, b, \alpha)$, of univalent harmonic mappings using multiplier transformation. We obtain a sufficient condition for harmonic univalent functions to be in $HS(k,\lambda,b,\alpha)$ and we prove that this condition is also necessary for the functions in the class $\overline{HS} (k,\lambda,b,\alpha)$. We also obtain extreme points, distortion bounds, convex combination, radius of convexity and Bernandi-Libera-Livingston integral for the functions in the class $\overline{HS}(k,\lambda,b,\alpha)$.

On a Subclass of Univalent Harmonic Mappings Convex in the Imaginary Direction

2020 ◽  
pp. 445-454
Author(s):  
Deepali Khurana ◽  
Sushma Gupta ◽  
Sukhjit Singh
Keyword(s):  
Present Article ◽  
Unit Disk ◽  
Imaginary Axis ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Distortion Bounds ◽  
Univalent Harmonic Mappings

In the present article, we consider a class of univalent harmonic mappings, $\mathcal{C}_{T} = \left\{ T_{c}[f] =\frac{f+czf'}{1+c}+\overline{\frac{f-czf'}{1+c}}; \; c>0\;\right\}$ and $f$ is convex univalent in $\mathbb{D}$, whose functions map the open unit disk $\mathbb{D}$ onto a domain convex in the direction of the imaginary axis. We estimate coefficient, growth and distortion bounds for the functions of the same class.

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