Class Of Harmonic Functions Research Articles

Bohr Inequalities for Certain Classes of Harmonic Mappings

Bohr Inequalities for Certain Classes of Harmonic Mappings

Harmonic functions associated with Pascal distribution series

The primary objective of this paper is to explore the application of a specific convolution operator, which incorporates the Pascal distribution series. Through this investigation, we establish essential inclusion relations between the harmonic class HΥ(ϱ,ξ) and other classes of harmonic functions, namely, harmonic starlike functions and harmonic convex functions, all defined within the confines of the open unit disk. Moreover, we derive consequential outcomes from our main findings. The study delves into the intriguing interplay among these harmonic function classes, shedding light on their underlying properties and mutual connections. The presented results contribute to a deeper understanding of harmonic functions and provide a valuable basis for further research in this area.

On the order of convolution consistence of certain classes of harmonic functions with varying arguments

"Making use of a modi ed Hadamard product or convolution of harmonic functions with varying arguments, combined with an integral operator, we study when these functions belong to a given class. Following an idea of U. Bednarz and J. Sokol we de ne the order of convolution consistence of three classes of functions and determine it for certain classes of harmonic functions with varying arguments de ned using a convolution operator."

Globally Existing Solutions to the Problem of Dirichlet for the Fractional 3D Poisson Equation

A general approach to solving the Dirichlet problem, both for bounded 3D domains and for their unbounded complements, in terms of the fractional (3D) Poisson equation, is presented. Lauren Schwartz class solutions are sought for tempered distributions. The solutions found are represented by a formula that contains the volume Riesz potential and the one-layer potential, the latter depending on the boundary data. Infinite regularity of fractional harmonic functions, analogous to the infinite smoothness of the classical harmonic functions, is also proved in the respective domain, no matter what the boundary conditions are. Other properties of the solutions, that are presumably of interest to mathematical physics, are also investigated. In particular, an intrinsic decay property, valid far from the common boundary, is shown.

The sharp refined Bohr–Rogosinski inequalities for certain classes of harmonic mappings

A class F consisting of analytic functions f ( z ) = ∑ n = 0 ∞ a n z n in the unit disc D = { z ∈ C : | z | < 1 } satisfies a Bohr phenomenon if there exists an 0 $ ]]> r f > 0 such that ∑ n = 1 ∞ | a n | r n ≤ d ( f ( 0 ) , ∂f ( D ) ) for every function f ∈ F , and | z | = r ≤ r f . The largest radius r f is the Bohr radius and the inequality ∑ n = 1 ∞ | a n | r n ≤ d ( f ( 0 ) , ∂f ( D ) ) is Bohr inequality for the class F , where ‘d’ is the Euclidean distance. In this paper, we prove sharp refinement of the Bohr–Rogosinski inequality for certain classes of harmonic mappings.

Bohr phenomenon for certain classes of harmonic mappings

Bohr phenomenon for analytic functions f, where f(z)=∑n=0∞anzn, was first introduced by Harald Bohr in 1914 and deals with finding the largest radius rf, 0<rf<1, such that the inequality ∑n=0∞|an||z|n<1 holds for |z|=r≤rf whenever |f(z)|<1 holds in the unit disk 𝔻={z∈ℂ:|z|<1}. The Bohr phenomenon for harmonic functions of the form f(z)=h+ḡ, where h(z)=∑n=0∞anzn and g(z)=∑n=1∞bnzn, is to find the largest radius rf, 0<rf<1 such that ∑n=1∞(|an|+|bn|)|z|n≤d(f(0),∂f(𝔻)) holds for |z|≤rf, where d(f(0),∂f(𝔻)) is the Euclidean distance between f(0) and the boundary of f(𝔻). We prove several improved versions of the sharp Bohr radius for the classes of harmonic and univalent functions. Further, we prove several corollaries as a consequence of the main results.

Growth of harmonic functions on biregular trees

On a biregular tree of degrees $q+1$ and $r+1$, we study the growth of two classes of harmonic functions. First, we prove that if $f$ is a bounded harmonic function on the tree and $x$, $y$ are two adjacent vertices, then $|f(x)-f(y)|\leq 2 (qr-1)\|f\|_\infty/((q+1)(r+1))$, thus generalizing a result of Cohen and Colonna for regular trees. Next, we prove that if $f$ is a positive harmonic function on the tree and $x$, $y$ are two vertices with $d(x,y)=2$, then $f(x)/(qr)\leq f(y)\leq qr\cdot f(x)$.

Some classes of harmonic functions on vector bundles

Some classes of harmonic functions on vector bundles

New Class of Close‐to‐Convex Harmonic Functions Defined by a Fourth‐Order Differential Inequality

In the recent past, various new subclasses of normalized harmonic functions have been defined in open unit disk U which satisfy second‐order and third‐order differential inequalities. Here, in this study, we define a new class of normalized harmonic functions in open unit disk U which is satisfying a fourth‐order differential inequality. We investigate some useful results such as close‐to‐convexity, coefficient bounds, growth estimates, sufficient coefficient condition, and convolution for the functions belonging to this new class of harmonic functions. In addition, under convex combination and convolution of its members, we prove that this new class is closed, and we also give some lemmas to prove our main results.

The fractional Malmheden theorem

&lt;abstract&gt;&lt;p&gt;We provide a fractional counterpart of the classical results by Schwarz and Malmheden on harmonic functions. From that we obtain a representation formula for $ s $-harmonic functions as a linear superposition of weighted classical harmonic functions which also entails a new proof of the fractional Harnack inequality. This proof also leads to optimal constants for the fractional Harnack inequality in the ball.&lt;/p&gt;&lt;/abstract&gt;

Bohr phenomenon for certain subclasses of harmonic mappings

The Bohr phenomenon for analytic functions of the form f(z)=∑n=0∞anzn, first introduced by Harald Bohr in 1914, deals with finding the largest radius rf, 0<rf<1, such that the inequality ∑n=0∞|anzn|≤1 holds whenever the inequality |f(z)|≤1 holds in the unit disk D={z∈C:|z|<1}. The exact value of this largest radius known as Bohr radius, which has been established to be rf=1/3. The Bohr phenomenon [1] for harmonic functions f of the form f(z)=h(z)+g(z)‾, where h(z)=∑n=0∞anzn and g(z)=∑n=1∞bnzn is to find the largest radius rf, 0<rf<1 such that∑n=1∞(|an|+|bn|)|z|n≤d(f(0),∂f(D)) holds for |z|≤rf, here d(f(0),∂f(D)) denotes the Euclidean distance between f(0) and the boundary of f(D). In this paper, we investigate the Bohr radius for several classes of harmonic functions in the unit disk D.

Classes of Harmonic Functions Defined By the Extended SăLăgean Operator

Classes of Harmonic Functions Defined By the Extended SăLăgean Operator

Bohr-Type Inequalities for Harmonic Mappings with a Multiple Zero at the Origin

In this paper, we first determine Bohr’s inequality for the class of harmonic mappings \(f=h+\overline{g}\) in the unit disk \(\mathbb {D}\), where either both \(h(z)=\sum _{n=0}^{\infty }a_{pn+m}z^{pn+m}\) and \(g(z)=\sum _{n=0}^{\infty }b_{pn+m}z^{pn+m}\) are analytic and bounded in \(\mathbb {D}\), or satisfies the condition \(|g'(z)|\le d|h'(z)|\) in \(\mathbb {D}\backslash \{0\}\) for some \(d\in [0,1]\) and h is bounded. In particular, we obtain Bohr’s inequality for the class of harmonic p-symmetric mappings. Also, we investigate the Bohr-type inequalities of harmonic mappings with a multiple zero at the origin and that most of results are proved to be sharp.

Further results on modified harmonic functions in three dimensions

The Weinstein equation , with , considered in , is a modification of the classical Laplace equation . Its solutions are called k‐modified harmonic functions. Whereas for positive integers k the Weinstein equation is relatively well understood, little is known if the parameter k is negative.The main result of this article is the statement that in case the negative integers are even, i.e., , we still have a Fischer‐type decomposition. For , the classical harmonic functions, this decomposition is well known. But also in case , a Fischer‐type decomposition holds true, a Fischer‐type decomposition holds true. Surprisingly in case or and probably in all higher negative odd cases, the decomposition doesn't hold.In case , we give a complete description of the vector space of homogeneous k‐modified harmonic polynomials of degree n in . Such a result is also at hand in case . Finally, in case of the classical harmonic functions, we give a description of the vector space .

Classes of harmonic functions defined by extended Sălăgean operator

UDC 517.57 The object of the present paper is to investigate classes of harmonic functions defined by the extended Sălăgea operator. By using the extreme points theory we obtain coefficients estimates and distortion theorems for these classes of functions. Some integral mean inequalities are also pointed out.

Classes of harmonic functions in 2D generalized Poincaré geometry

By using the additive and multiplicative separation of variables we find some classes of solutions of the Laplace equation for a generalization of the Poincar? upper half plane metric. Non-constant totally geodesic functions implies the flat metric and several examples are studied including the Hamilton?s cigar Ricci soliton. The Bochner formula is discussed for our generalized Poincar? metric and for its important particular cases.

Harmonic Function with Correlated Coefficients

In the paper we introduce an idea of harmonic functions with correlated coefficients which generalize the ideas of harmonic functions with negative coefficients introduced by Silverman and harmonic functions with varying coefficients defined by Jahangiri and Silverman. Next we define classes of harmonic functions with correlated coefficients in terms of generalized Dziok-Srivastava operator. By using extreme points theory, we obtain estimations of classical convex functionals on the defined classes of functions. Some applications of the main results are also considered.

Norm of the pre-Schwarzian derivative, Bloch's constant and coefficient bounds in some classes of harmonic mappings

The aim of the paper is a study of a family of sense-preserving complex valued harmonic functions f that are normalized in the open unit disk, and such that its analytic part is convex in one direction. We prove the univalence and establish estimates on pre-Schwarzian derivative and the Bloch constant for co-analytic part of harmonic mapping. The bounds of the coefficients of co-analytic part are also given.

Some classes of harmonic mappings with analytic part defined by subordination

Let S H role=presentation> S H S H S_{H} be the class of functions f = h + g ¯ role=presentation> f = h + g ¯ f = h + g ¯ f=h+\bar{g} that are harmonic univalent and sense-preserving in the open unit disk U = { z ∈ C : | z | 𝕌 = { z ∈ ℂ : | z | h , h , h, g role=presentation> g g g are analytic and f ( 0 ) = f z ′ ( 0 ) − 1 = 0 role=presentation> f ( 0 ) = f ′ z ( 0 ) − 1 = 0 f ( 0 ) = f z ′ ( 0 ) − 1 = 0 f(0)=f_{z}'(0)-1=0 in U . role=presentation> 𝕌 . U . \mathbb{U}. In this paper, we investigate the properties of some subclasses of S H role=presentation> S H S H S_{H} such that h ( z ) role=presentation> h ( z ) h ( z ) h(z) is a starlike (or convex) function defined by subordination. We provide coefficient estimates, extremal function, distortion and growth estimates of g role=presentation> g g g , growth, and Jacobian estimates of f role=presentation> f f f . We also obtain area estimates and covering theorems of the classes. The results presented here generalize some known results.

Schwarz Pick type inequalities for harmonic maps between Riemann surfaces

ABSTRACTWe study the Schwarz lemma for harmonic mappings in general setting and prove among the other results that: if is the class of harmonic mappings f of the unit disk into the geodesic disk of a Riemann surface , which is disjoint from its cut-locus with radius M satisfying some curvature restrictions, then there is a locally Lipschitz homeomorphism , convex in , with and such that for with we have for.