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.

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

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).

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 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.

scholarly journals The Generalized Janowski Starlike and Close-to-Starlike Log-Harmonic Mappings

2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Maisarah Haji Mohd ◽  
Maslina Darus
Keyword(s):  
Unit Disc ◽  
Harmonic Mapping ◽  
Starlike Functions ◽  
Starlike Function ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disc

Motivated by the success of the Janowski starlike function, we consider here closely related functions for log-harmonic mappings of the form defined on the open unit disc . The functions are in the class of the generalized Janowski starlike log-harmonic mapping, , with the functional in the class of the generalized Janowski starlike functions, . By means of these functions, we obtained results on the generalized Janowski close-to-starlike log-harmonic mappings, .

scholarly journals Quasiconformal harmonic mappings and close-to-convex domains

Filomat ◽  
2010 ◽  
Vol 24 (1) ◽  
pp. 63-68 ◽  
Author(s):  
David Kalaj
Keyword(s):  
Unit Disk ◽  
Convex Domain ◽  
Harmonic Mapping ◽  
Harmonic Mappings ◽  
Convex Domains ◽  
Primary 30C55 ◽  
Subject Classifications ◽  
Mathematics Subject Classifications ◽  
Secondary 31C05

Let f = h + ? be a univalent sense preserving harmonic mapping of the unit disk U onto a convex domain ?. It is proved that: for every a such that |a| < 1 (resp. |a| = 1) the mapping ?a = h + a? is an |a| quasiconformal (a univalent) close-to-convex harmonic mapping. This gives an answer to a question posed by Chuaqui and Hern?ndez (J. Math. Anal. Appl. (2007)). 2010 Mathematics Subject Classifications. Primary 30C55, Secondary 31C05. .

Linear combinations of a class of harmonic univalent mappings

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3111-3121
Author(s):  
Bo-Yong Long ◽  
Michael Dorff
Keyword(s):  
Linear Combination ◽  
Imaginary Axis ◽  
Sufficient Conditions ◽  
Harmonic Mapping ◽  
Harmonic Mappings ◽  
Linear Combinations ◽  
Vertical Strip ◽  
Planar Harmonic Mapping ◽  
Complex Valued ◽  
Planar Harmonic Mappings

A planar harmonic mapping is a complex-valued function f : U ? C of the form f (x + iy) = u(x,y) + iv(x,y), where u and v are both real harmonic. Such a function can be written as f = h + g?, where h and g are both analytic; the function ? = g'=h' is called the dilatation of f. We consider the linear combinations of planar harmonic mappings that are the vertical shears of the asymmetrical vertical strip mappings j(z) = 1/2isin?j log (1+zei?j/ 1+ze-i?j) with various dilatations, where ?j ? [?/2,?), j=1,2. We prove sufficient conditions for the linear combination of this class of harmonic univalent mappings to be univalent and convex in the direction of the imaginary axis.

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