scholarly journals A series expansion for generalized harmonic functions

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Markus Klintborg ◽  
Anders Olofsson
Keyword(s):  
Harmonic Function ◽  
Series Expansion ◽  
Unit Disc ◽  
Harmonic Functions ◽  
Poisson Kernel ◽  
Open Unit ◽  
Open Unit Disc ◽  
Generalized Harmonic Function ◽  
Generalized Harmonic Functions

AbstractWe consider a class of generalized harmonic functions in the open unit disc in the complex plane. Our main results concern a canonical series expansion for such functions. Of particular interest is a certain individual generalized harmonic function which suitably normalized plays the role of an associated Poisson kernel.

scholarly journals Domination of the supremum of a bounded harmonic function by its supremum over a countable subset

1987 ◽  
Vol 30 (3) ◽  
pp. 471-477 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Harmonic Function ◽  
Unit Disc ◽  
Harmonic Functions ◽  
Open Unit ◽  
Countable Subset ◽  
Open Unit Disc ◽  
Image Position ◽  
Bounded Harmonic Functions ◽  
Bounded Harmonic Function

For what sequences {an} of points of the open unit disc D does there exist a constant k such thatfor all bounded harmonic functions f on D?

scholarly journals Mapping properties of Janowski-type harmonic functions involving Mittag-Leffler function

2021 ◽  
Vol 6 (12) ◽  
pp. 13235-13246
Author(s):  
Murugusundaramoorthy Gangadharan ◽  
◽  
Vijaya Kaliyappan ◽  
Hijaz Ahmad ◽  
K. H. Mahmoud ◽  
...  
Keyword(s):  
Convolution Operator ◽  
Unit Disc ◽  
Harmonic Functions ◽  
Univalent Functions ◽  
Open Unit ◽  
Open Unit Disc

<abstract><p>In this paper, we examine a connotation between certain subclasses of harmonic univalent functions by applying certain convolution operator regarding Mittag-Leffler function. To be more precise, we confer such influences with Janowski-type harmonic univalent functions in the open unit disc $ \mathbb{D}. $</p></abstract>

scholarly journals Vanishing l1-sums of the Poisson kernel, and sums with positive coefficients

1989 ◽  
Vol 32 (3) ◽  
pp. 431-447 ◽  
Author(s):  
F. F. Bonsall ◽  
D. Walsh
Keyword(s):  
Unit Disc ◽  
Poisson Kernel ◽  
Open Unit ◽  
Complex Numbers ◽  
Open Unit Disc ◽  
Image Position ◽  
Positive Coefficients

For z in D and ζ in ∂D, we denote by pz(ζ) the Poisson kernel (1 − │z│2)│1 − z̄ζ−2 for the open unit disc D. We ask for what countable sets {an:n∈ℕ} of points of D there exist complex numbers λn withby which we mean that the series converges to zero in the norm of L1(∂D).

scholarly journals Properties of harmonic functions which are convex of order $ \bf \beta $ with respect to symmetric points

2009 ◽  
Vol 40 (1) ◽  
pp. 31-39 ◽  
Author(s):  
Aini Janteng ◽  
Suzeini Abdul Halim
Keyword(s):  
Unit Disc ◽  
Harmonic Functions ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disc ◽  
Symmetric Points ◽  
Complex Valued ◽  
Class Of Functions

Let $ \mathcal{H} $ denote the class of functions $ f $ which are harmonic and univalent in the open unit disc $ {D=\{z:|z|<1\}} $. This paper defines and investigates a family of complex-valued harmonic functions that are orientation preserving and univalent in $ \mathcal{D} $ and are related to the functions convex of order $ \beta(0\leq \beta <1) $, with respect to symmetric points. We obtain coefficient conditions, growth result, extreme points, convolution and convex combinations for the above harmonic functions.

scholarly journals Subclass of Multivalent Harmonic Functions with Missing Coefficients

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
R. M. El-Ashwah
Keyword(s):  
Unit Disc ◽  
Harmonic Functions ◽  
Extreme Points ◽  
Distortion Theorem ◽  
Open Unit ◽  
Open Unit Disc ◽  
Basic Properties

We have studied subclass of multivalent harmonic functions with missing coefficients in the open unit disc and obtained the basic properties such as coefficient characterization and distortion theorem, extreme points, and convolution.

scholarly journals A subclass of harmonic univalent functions with positive coefficients

2010 ◽  
Vol 41 (3) ◽  
pp. 261-269 ◽  
Author(s):  
K. K. Dixit ◽  
Saurabh Porwal
Keyword(s):  
Integral Operator ◽  
Unit Disc ◽  
Harmonic Functions ◽  
Univalent Functions ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disc ◽  
Distortion Bounds ◽  
Positive Coefficients ◽  
Complex Valued

Complex-valued harmonic functions that are univalent and sense-preserving in the open unit disc $U$ can be written in the form $f=h+\bar g$, where $h$ and $g$ are analytic in $U$. In this paper authors introduce the class, $R_H(\beta)$, $(1<\beta \le 2)$ consisting of harmonic univalent functions $f=h+\bar g$, where $h$ and $g$ are of the form $ h(z)=z+ \sum_{k=2}^\infty |a_k|z^k $ and $ g(z)= \sum_{k=1}^\infty |b_k| z^k $ for which $\Re\{h'(z)+g'(z)\}<\beta$. We obtain distortion bounds extreme points and radii of convexity for functions belonging to this class and discuss a class  preserving integral operator. We also show that class studied in this paper is closed under convolution and convex combinations.

scholarly journals A CHARACTERIZATION OF THE HARMONIC BLOCH SPACE AND THE HARMONIC BESOV SPACES BY AN OSCILLATION

2002 ◽  
Vol 45 (1) ◽  
pp. 229-239 ◽  
Author(s):  
Rikio Yoneda
Keyword(s):  
Besov Spaces ◽  
Unit Disc ◽  
Harmonic Functions ◽  
Bloch Space ◽  
Subject Classification ◽  
Open Unit ◽  
Open Unit Disc ◽  
Primary 46E15 ◽  
Alpha Beta

AbstractWe characterize the Bloch space and the Besov spaces of harmonic functions on the open unit disc $D$ by using the following oscillation:$$ \sup_\{\beta(z,w)\ltr\}(1-|z|^2)^{\alpha}(1-|w|^2)^{\beta}\biggl|\frac{\hat{D}^{(n-1)}h(z)-\hat{D}^{(n-1)}h(w)}{z-w}\biggr|, $$where $\alpha+\beta=n$, $\alpha,\beta\in\mathbb{R}$ and $\displaystyle{\hat{D}^{(n)}=(\partial^{n}/\partial^{n}z+\partial^{n}/\partial^{n}\bar{z})}$.AMS 2000 Mathematics subject classification: Primary 46E15

An argument of a function in H1/2

2012 ◽  
Vol 55 (2) ◽  
pp. 507-511
Author(s):  
Takahiko Nakazi ◽  
Takanori Yamamoto
Keyword(s):  
Hardy Space ◽  
Simple Proof ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc

AbstractLet H1/2 be the Hardy space on the open unit disc. For two non-zero functions f and g in H1/2, we study the relation between f and g when f/g ≥ 0 a.e. on ∂D. Then we generalize a theorem of Neuwirth and Newman and Helson and Sarason with a simple proof.

scholarly journals Harmonic univalent functions defined by post quantum calculus operators

2019 ◽  
Vol 11 (1) ◽  
pp. 5-17 ◽  
Author(s):  
Om P. Ahuja ◽  
Asena Çetinkaya ◽  
V. Ravichandran
Keyword(s):  
Unit Disc ◽  
Univalent Functions ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disc ◽  
Quantum Calculus ◽  
Special Cases ◽  
The Family ◽  
Covering Theorems

Abstract We study a family of harmonic univalent functions in the open unit disc defined by using post quantum calculus operators. We first obtained a coefficient characterization of these functions. Using this, coefficients estimates, distortion and covering theorems were also obtained. The extreme points of the family and a radius result were also obtained. The results obtained include several known results as special cases.

Some properties of the analytic functions with bounded radius rotation

2019 ◽  
Vol 28 (1) ◽  
pp. 85-90
Author(s):  
YASAR POLATOGLU ◽  
◽  
ASENA CETINKAYA ◽  
OYA MERT ◽  
◽  
...  
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Starlike Functions ◽  
Open Unit ◽  
Coefficient Estimate ◽  
Open Unit Disc ◽  
Radius Of Starlikeness ◽  
Growth Theorem ◽  
Bounded Radius Rotation

In the present paper, we introduce a new subclass of normalized analytic starlike functions by using bounded radius rotation associated with q- analogues in the open unit disc \mathbb D. We investigate growth theorem, radius of starlikeness and coefficient estimate for the new subclass of starlike functions by using bounded radius rotation associated with q- analogues denoted by \mathcal{R}_k(q), where k\geq2, q\in(0,1).

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