Applying the argument principle to count zeros of harmonic functions with poles

Harmonic functions are a classic title in the class of geometric functions. Many researchers have studied these function classes from past to present, and since it has a wide range of applications, it is still a popular class. In this study, we will examine harmonic univalent functions, a subclass of harmonic functions. In this study, a subclass of harmonic univalent functions will be examined. Let H denote the class of continuous complex-valued harmonic functions which are harmonic in the open unit disk U={z ϵ C∶|z| |g'(z)| (see [3]). Throughout this paper, we will use introductory notations and delineations of the (p, q)- calculus. The aim of the present paper is to find connections between (p,q)-starlike harmonic univalent functions involving (p,q)-Poisson distribution series.

A complex-valued harmonic functions that are univalent and sense preserving in the unit disk U can be written in the form $$f = h + \bar g$$ , where h and g are analytic in U. We define and investigate a new class SHPλ(α, β) by generalized Salagean operator of harmonic univalent functions. We give sufficient coefficient conditions for normalized harmonic functions in the class SHPλ(α,β). These conditions are also shown to be necessary when the coefficients are negative. This leads to distortion bounds and extreme points.

A subclass of harmonic univalent functions with negative coefficients

In this chapter the notation is made simpler by considering harmonic functions which are complex valued functions and have real and imaginary parts satisfying the Laplace equation in \(N\)-variables. The Dirichlet problem is to find such a function of \(N\) variables, satisfying the Laplace equation in the interior of a domain \(D\) and being equal to a given continuous complex-valued function \(g\) on the boundary \(\Gamma \) of \(D\). The case of interest in applications is \(N\)=3, but there is no essential difference in considering general \(N\). In addition to specifying that the bounded open set \(D\) not have any holes, bubbles in three dimensions, which was enough in two dimensions, additional conditions are necessary in three or higher dimensions, and the relevant Poincare condition is discussed. The proof uses the Hahn-Banach Theorem applied in the space \(C(K)\) of complex-valued continuous functions defined on the boundary K of the domain. In the final result it is shown that there is an approximate solution using any one chosen function \(f(z)\), which is not a polynomial and which has a power series convergent (about z = 0). This approximate solution is a finite sum of the form \({c_n}f({a^{(n)}} \cdot x + {b^n} \cdot x + {d_n})\) where \({c_n}\) and \({d_n}\) are complex numbers, \(a^{(n)}\) and \(b^n\) are vectors in \(R^N\) with the same length, ||\({a^{(n)}}\)|| = ||\({b^{(n)}}\)||, which are perpendicular \(a^{(n)}\) \(\cdot \) \(b^n\) = 0, and x is a point in \(R^N\). The proof relies on a lemma which shows how to construct a harmonic function in \(N\) variables from a harmonic function in two variables, which will be applied to the real and imaginary parts of \(f(z)\), and a representation theorem for the space of harmonic polynomials in \(N\)-variables which are homogeneous of degree \(m\). Note that this result demonstrates the ability of a function of two variables, \(f(z)\), to generate an approximate solution of the Dirichlet problem in \(N\)-variables.

Certain multipliers of univalent harmonic functions

The argument principle is an important and useful result for meromorphic functions on domains in the plane. Duren, Hengartner and Laugesen have obtained an argument principle for harmonic mappings on Jordan domains. This paper is concerned with proving an argument principle for harmonic mappings that have isolated singularities. Examples are included to show how the geometric behavior of harmonic mappings differs from the meromorphic case. Also included in the paper is a definition of “pole” at an isolated singular point of a harmonic mapping and a “partitioning theorem” for the image space that yields components whose values are assumed the same number of times in the appropriate region.

We prove several global univalence theorems for locally invertible harmonic mappings with certain prescribed boundary behavior in simply and multiply connected domains. In particular, we consider mappings with singular boundary points when the argument principle is not applicable. In addition, we examine harmonic mappings connected with the famous von Mises coordinates. It is shown the univalence of every locally univalent von Mises harmonic mapping defined in a domain convex in the direction of ordinate axis.

Complex-valued harmonic functions that are univalent and sense preserving in the open unit disk can be written in the form $f=h+\overline{g}$ , where h and g are analytic. In this paper we investigate some classes of univalent harmonic functions with varying coefficients related to Janowski functions. By using the extreme points theory we obtain necessary and sufficient convolution conditions, coefficients estimates, distortion theorems, and integral mean inequalities for these classes of functions. The radii of starlikeness and convexity for these classes are also determined.

Harmonic Functions Starlike in the Unit Disk

Complex-valued harmonic functions that are univalent and sense-preserving in the open unit disk Δ can be written in the form f = h+g¯, where h and g are analytic in Δ several mathematicians examined classes of complex harmonic functions in the unit disk with some coefficient conditions. In this paper we introduce some results which generalize problems considered in other previous papers. The main results concern starlikeness and convexity of harmonic functions satisfying a three parameter coefficient condition.

We derive a formula for the number of pre-images under a non-degenerate\nharmonic mapping $f$, using the argument principle. This formula reveals a\nconnection between the pre-images and the caustics. Our results allow to deduce\nthe number of pre-images under $f$ geometrically for every non-caustic point.\nWe approximately locate the pre-images of points near the caustics. Moreover,\nwe apply our results to prove that for every $k = n, n+1, \\ldots, n^2$ there\nexists a harmonic polynomial of degree $n$ with $k$ zeros.\n

A Guide to Complex Variables gives the reader a quick and accessible introduction to the key topics. While the coverage is not comprehensive, it certainly gives the reader a solid grounding in this fundamental area. There are many figures and examples to illustrate the principal ideas, and the exposition is lively and inviting. An undergraduate wanting to have a first look at this subject or a graduate student preparing for the qualifying exams, will find this book to be a useful resource. In addition to important ideas from the Cauchy theory, the book also includes the Riemann mapping theorem, harmonic functions, the argument principle, general conformal mapping and dozens of other central topics. Readers will find this book to be a useful companion to more exhaustive texts in the field. It is a valuable resource for mathematicians and non-mathematicians alike. Steven Krantz is well-known for his skill in expository writing and this volume confirms it. He is the author of more than 50 books, and more than 150 scholarly papers. The MAA has awarded him both the Beckenbach Book Prize and the Chauvenet Prize.

A classical result of Hardy and Littlewood says that if $$f=u+iv$$ f = u + i v is analytic in the unit disk $${\mathbb {D}}$$ D and u is in the harmonic Bergman space $$a^p$$ a p ( $$0<p<\infty $$ 0 < p < ∞ ), then v is also in $$a^p$$ a p . This complements a celebrated result of M. Riesz on Hardy spaces, which only holds for $$1<p<\infty $$ 1 < p < ∞ . These results do not extend directly to complex-valued harmonic functions. We prove that the Hardy-Littlewood theorem holds for a harmonic function $$f=u+iv$$ f = u + i v if we place the assumption that f is quasiregular in $${\mathbb {D}}$$ D . This makes further progress on the recent Riesz type theorems for harmonic quasiregular mappings by several authors. Then we consider univalent harmonic mappings in $${\mathbb {D}}$$ D and study their membership in Bergman spaces. In particular, we produce a non-trivial range of $$p>0$$ p > 0 such that every univalent harmonic function f (and the partial derivatives $$f_\theta ,\, rf_r$$ f θ , r f r ) is of class $$a^p$$ a p . This result extends nicely to harmonic quasiconformal mappings in $${\mathbb {D}}$$ D .

In this paper, we introduce a new generalized Noor-type operator of harmonic p-valent functions associated with the Fox-Wright generalized hypergeometric functions (FWGH-functions). Furthermore, we consider a new subclass of complex-valued harmonic multivalent functions based on this new operator. Several geometric properties for this subclass are also discussed.

This article examines the basic concept of a μ-regular function compared to an analytic function, in connection with the fact that these two complex functions are built from panharmonic functions and harmonic functions, which are solutions of the Yukawa equation and the Laplace partial differential equation, respectively. The two differential equations differ only in that the Yukawa equation has a positive constant on the right-hand side, while the Laplace equation has a zero constant there. In general, the Laplace equation is a special case of the Yukawa equation. Similarly, the complex-valued functions constructed from these two real-valued functions, namely, harmonic and panharmonic functions are considered. Analytic functions are those whose real and imaginary parts are harmonic functions and which satisfy the Cauchy-Riemann equations, whereas μ-regular functions are complex functions whose real and imaginary parts are panharmonic functions and satisfy the generalized Cauchy-Riemann equations. In fact, if in the generalized Cauchy-Riemann equations, then the standard Cauchy-Riemann equations are recovered.