Bloch constant and Landau's theorem for planar p-harmonic mappings

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.

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RADII OF HARMONIC MAPPINGS IN THE PLANE

Journal of the Australian Mathematical Society ◽

10.1017/s1446788716000252 ◽

2016 ◽

Vol 102 (3) ◽

pp. 331-347 ◽

Cited By ~ 3

Author(s):

BO-YONG LONG ◽

HUA-YING HUANG

Keyword(s):

Convex Combination ◽

Harmonic Mappings ◽

Bloch Constant

In this paper, for the convolution and convex combination of harmonic mappings, the radii of univalence, full convexity and starlikeness of order $\unicode[STIX]{x1D6FC}$ are explored. All results are sharp. By way of application, the univalent radius and the Bloch constant of the convolution of two bounded harmonic mappings are obtained.

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

Mathematical Modelling and Analysis ◽

10.3846/13926292.2013.756834 ◽

2013 ◽

Vol 18 (1) ◽

pp. 66-79 ◽

Cited By ~ 5

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.

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A Class of Harmonic Starlike Functions defined by Multiplier Transformation

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.1.36 ◽

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 $\overline{HS}(k,\lambda,b,\alpha)$.

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

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.1.35 ◽

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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