scholarly journals Heinz-Schwarz inequalities for harmonic mappings in the unit ball

2016 ◽  
Vol 41 ◽  
pp. 457-464 ◽  
Author(s):  
Kalaj David
Keyword(s):  
Unit Ball ◽  
Harmonic Mappings

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 ON HARMONIC BLOCH SPACES IN THE UNIT BALL OF ℂn

2011 ◽  
Vol 84 (1) ◽  
pp. 67-78 ◽  
Author(s):  
SH. CHEN ◽  
X. WANG
Keyword(s):  
Unit Ball ◽  
Lipschitz Functions ◽  
Harmonic Mappings ◽  
Bloch Spaces ◽  
Bloch Constant ◽  
Vector Valued

AbstractIn this paper, our main aim is to discuss the properties of harmonic mappings in the unit ball 𝔹n. First, we characterize the harmonic Bloch spaces and the little harmonic Bloch spaces from 𝔹n to ℂ in terms of weighted Lipschitz functions. Then we prove the existence of a Landau–Bloch constant for a class of vector-valued harmonic Bloch mappings from 𝔹n to ℂn.

scholarly journals WEIGHTED LIPSCHITZ CONTINUITY, SCHWARZ–PICK'S LEMMA AND LANDAU–BLOCH'S THEOREM FOR HYPERBOLIC-HARMONIC MAPPINGS IN ℂN

2013 ◽  
Vol 18 (1) ◽  
pp. 66-79 ◽  
Author(s):  
Shaolin Chen ◽  
Saminathan Ponnusamy ◽  
Xiantao Wang
Keyword(s):  
Unit Ball ◽  
Harmonic Functions ◽  
Lipschitz Continuity ◽  
Type Theorem ◽  
Harmonic Mappings ◽  
Bloch Spaces ◽  
Bloch Constant ◽  
Bloch’S Theorem ◽  
Hyperbolic Harmonic Functions ◽  
The Relationship

In this paper, we discuss some properties on hyperbolic-harmonic functions in the unit ball of ℂ n . First, we investigate the relationship between the weighted Lipschitz functions and the hyperbolic-harmonic Bloch spaces. Then we establish the Schwarz–Pick type theorem for hyperbolic-harmonic functions and apply it to prove the existence of Landau-Bloch constant for functions in α-Bloch spaces.

scholarly journals On conformal, harmonic mappings and Dirichlet’s integral

Filomat ◽  
2011 ◽  
Vol 25 (2) ◽  
pp. 91-108
Author(s):  
David Kalaj ◽  
Miodrag Mateljevic
Keyword(s):  
Sobolev Space ◽  
Unit Ball ◽  
Higher Dimensions ◽  
Harmonic Mappings ◽  
Dirichlet Integral ◽  
Dirichlet Integrals ◽  
Derivatives Of

This paper has an expository character, however we present as well some new results and new proofs. We prove a complex version of Dirichlet?s principle in the plane and give some applications of it as well as estimates of Dirichlet?s integral from below. Some of the results in the plane are generalized to higher dimensions. Roughly speaking, under the appropriate conditions we estimate the n-Dirichlet integral of a mapping u defined on a domain ? ? Rn , n ? 2, by the measure of u(?) and show that equality holds if and only if it is injective conformal. Also some sharp inequalities related to the L2 norms of the radial derivatives of vector harmonic mappings from the unit ball in Rn, n ? 2, are given. As an application, we estimate the 2-Dirichlet integrals of mappings in the Sobolev space Wi2.

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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