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.

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).

On a subclass of p-harmonic mappings

2013 ◽  
Vol 44 (3) ◽  
pp. 313-325 ◽  
Author(s):  
Saurabh Porwal ◽  
Kaushal Kishore Dixit
Keyword(s):  
Extreme Points ◽  
Harmonic Mapping ◽  
Harmonic Mappings

The purpose of the present paper is to introduce two new classes $HS_p(\alpha)$ and $HC_p(\alpha)$ of $p$-harmonic mappings together with their corresponding subclasses $HS^0_p(\alpha)$ and $HC^0_p(\alpha)$. We prove that the mappings in $HS_p(\alpha)$ and $HC_p(\alpha)$ are univalent and sense-preserving in $U$ and obtain extreme points of $HS^0_p(\alpha)$ and $HC^0_p(\alpha)$, $HS_p(\alpha)\cap T_p$ and $HC_p(\alpha)\cap T_p$ are determined, where $T_p$ denotes the set of $p$-harmonic mapping with non negative coefficients. Finally, we establish the existence of the neighborhoods of mappings in $HC_p(\alpha)$. Relevant connections of the results presented here with various known results are briefly indicated.

THE MODULUS OF THE IMAGE ANNULI UNDER UNIVALENT HARMONIC MAPPINGS AND A CONJECTURE OF NITSCHE

2001 ◽  
Vol 64 (2) ◽  
pp. 369-384 ◽  
Author(s):  
ABDALLAH LYZZAIK
Keyword(s):  
Upper Bound ◽  
Harmonic Mapping ◽  
Ring Domain ◽  
Harmonic Mappings ◽  
Univalent Harmonic Mappings

The object of the paper is to show that if f is a univalent, harmonic mapping of the annulus A(r, 1) = {z : r < [mid ]z[mid ] < 1} onto the annulus A(R, 1), and if s is the length of the segment of the Grötzsch ring domain associated with A(r, 1), then R < s. This gives the first, quantitative upper bound of R, which relates to a question of J. C. C. Nitsche that he raised in 1962. The question of whether this bound is sharp remains open.

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.

scholarly journals Quasiconformality of harmonic mappings between Jordan domains

Filomat ◽  
2010 ◽  
Vol 24 (3) ◽  
pp. 111-125 ◽  
Author(s):  
Miodrag Mateljevic ◽  
Vladimir Bozin ◽  
Miljan Knezevic
Keyword(s):  
Unit Disc ◽  
Necessary Conditions ◽  
Harmonic Mapping ◽  
Boundary Function ◽  
Harmonic Mappings ◽  
Existence Problem ◽  
Sufficient And Necessary Conditions

Suppose that h is a harmonic mapping of the unit disc onto a C 1,? domain D. We give sufficient and necessary conditions in terms of boundary function that h is q.c. We announce some new results and also outline application to existence problem of mean distortion minimizers in the Universal Teichm?ller space.

scholarly journals Quasiconformality of harmonic mappings between Jordan domains

Filomat ◽  
2012 ◽  
Vol 26 (3) ◽  
pp. 479-510 ◽  
Author(s):  
Miodrag Mateljevic
Keyword(s):  
Half Plane ◽  
Unit Disc ◽  
Necessary Conditions ◽  
Harmonic Mapping ◽  
Boundary Function ◽  
Harmonic Mappings ◽  
C1 Domain ◽  
Sufficient And Necessary Conditions ◽  
Upper Half Plane

Suppose that h is a harmonic mapping of the unit disc onto a C1, ? domain D. Then h is q.c. if and only if it is bi-Lipschitz. In particular, we consider sufficient and necessary conditions in terms of boundary function that h is q.c. We give a review of recent related results including the case when domain is the upper half plane. We also consider harmonic mapping with respect to ? metric on codomain.

Quasi-isometricity and equivalent moduli of continuity of planar 1/|ω|2-harmonic mappings

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 335-345
Author(s):  
Qi Yi ◽  
Shi Qingtian
Keyword(s):  
Quasiconformal Mapping ◽  
Connected Domain ◽  
Quasiregular Mapping ◽  
Harmonic Mappings ◽  
Moduli Of Continuity ◽  
Lipschitz Continuous ◽  
Quasihyperbolic Metric ◽  
Simply Connected Domain ◽  
Simply Connected

In this paper, we prove that 1/|?|2-harmonic quasiconformal mapping is bi-Lipschitz continuous with respect to quasihyperbolic metric on every proper domain of C\{0}. Hence, it is hyperbolic quasi-isometry in every simply connected domain on C\{0}, which generalized the result obtained in [14]. Meanwhile, the equivalent moduli of continuity for 1/|?|2-harmonic quasiregular mapping are discussed as a by-product.

scholarly journals The Properties of a New Subclass of Harmonic Univalent Mappings

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Zhi-Hong Liu ◽  
Ying-Chun Li
Keyword(s):  
Harmonic Functions ◽  
Horizontal Direction ◽  
Harmonic Mappings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

We introduced a new subclass of univalent harmonic functions defined by the shear construction in the present paper. First, we showed that the convolutions of two special subclass harmonic mappings are convex in the horizontal direction. Secondly, we proved a necessary and sufficient condition for the above subclass of harmonic mappings to be convex in the horizontal direction. We also presented some basic examples of univalent harmonic functions explaining the behavior of the image domains.

QUASICONFORMAL HARMONIC MAPPINGS BETWEEN DOMAINS CONTAINING INFINITY

2020 ◽  
Vol 102 (1) ◽  
pp. 109-117
Author(s):  
DAVID KALAJ
Keyword(s):  
Quasiconformal Mapping ◽  
Complex Plane ◽  
Riemannian Metrics ◽  
Harmonic Mappings ◽  
Lipschitz Continuous ◽  
Extended Complex ◽  
Extended Complex Plane

Assume that $\unicode[STIX]{x1D6FA}$ and $D$ are two domains with compact smooth boundaries in the extended complex plane $\overline{\mathbf{C}}$. We prove that every quasiconformal mapping between $\unicode[STIX]{x1D6FA}$ and $D$ mapping $\infty$ onto itself is bi-Lipschitz continuous with respect to both the Euclidean and Riemannian metrics.

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