Univalent harmonic mappings II

Albert E. Livingston

Displaying similar documents to “Univalent harmonic mappings II”

Landau's theorem for p-harmonic mappings in several variables

Sh. Chen, S. Ponnusamy, X. Wang (2012)

Annales Polonici Mathematici

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A 2p-times continuously differentiable complex-valued function f = u + iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation $Δ^{p} f = 0$ , where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂⁿ is a domain, then a function $f : Ω \to ℂ^{m}$ is said to be p-harmonic in Ω if each component function $f_{i}$ (i∈ 1,...,m) of $f = (f ₁, . . ., f_{m})$ is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch’s theorem for a class of p-harmonic mappings...

Estimates of the real part of $\frac{f (z)}{z}$ for some classes of univalent functions

M. Obradović (1984)

Matematički Vesnik

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Sakaguchi type functions defined by balancing polynomials

Gunasekar Saravanan, Sudharsanan Baskaran, Balasubramaniam Vanithakumari, Serap Bulut (2025)

Mathematica Bohemica

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The class of Sakaguchi type functions defined by balancing polynomials has been introduced as a novel subclass of bi-univalent functions. The bounds for the Fekete-Szegö inequality and the initial coefficients $| a_{2} |$ and $| a_{3} |$ have also been estimated.

Region of variability for functions with positive real part

Saminathan Ponnusamy, Allu Vasudevarao (2010)

Annales Polonici Mathematici

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For γ ∈ ℂ such that |γ| < π/2 and 0 ≤ β < 1, let $_{γ, β}$ denote the class of all analytic functions P in the unit disk with P(0) = 1 and $R e (e^{i γ} P (z)) > β c o s γ$ in . For any fixed z₀ ∈ and λ ∈ ̅, we shall determine the region of variability $V_{} (z ₀, λ)$ for $\int_{0}^{z ₀} P (ζ) d ζ$ when P ranges over the class $(λ) = P \in_{γ, β} : P^{'} (0) = 2 (1 - β) λ e^{- i γ} c o s γ .$ As a consequence, we present the region of variability for some subclasses of univalent functions. We also graphically illustrate the region of variability for several sets of parameters.

Initial Maclaurin coefficient estimates for $λ$ -pseudo-starlike bi-univalent functions associated with Sakaguchi-type functions

Abbas Kareem Wanas, Basem Aref Frasin (2022)

Mathematica Bohemica

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We introduce and study two certain classes of holomorphic and bi-univalent functions associating $λ$ -pseudo-starlike functions with Sakaguchi-type functions. We determine upper bounds for the Taylor–Maclaurin coefficients $| a_{2} |$ and $| a_{3} |$ for functions belonging to these classes. Further we point out certain special cases for our results.

Injectivity of sections of convex harmonic mappings and convolution theorems

Liulan Li, Saminathan Ponnusamy (2016)

Czechoslovak Mathematical Journal

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We consider the class $ℋ_{0}$ of sense-preserving harmonic functions $f = h + \bar{g}$ defined in the unit disk $| z | < 1$ and normalized so that $h (0) = 0 = h^{'} (0) - 1$ and $g (0) = 0 = g^{'} (0)$ , where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes $𝒫_{H}^{0} (α)$ and $𝒢_{H}^{0} (β)$ of functions from $ℋ_{0}$ and show that if $f \in 𝒫_{H}^{0} (α)$ and $F \in 𝒢_{H}^{0} (β)$ , then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters $α$ and $β$ are satisfied. In the second part we study the harmonic sections...

Sharp estimation of the coefficients of bounded univalent functions close to identity

Lucjan Siewierski

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CONTENTSIntroduction...............................................................................................................................................................................5Definitions and notation.........................................................................................................................................................7The main result........................................................................................................................................................................91....

On the class of functions defined in a halfplane and starlike with respect to the boundary point

Adam Lecko (2002)

Annales Polonici Mathematici

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The purpose of this paper is to study the class $*_{\infty} (ℍ)$ of univalent analytic functions defined in the right halfplane ℍ and starlike w.r.t. the boundary point at infinity. An analytic characterization of functions in $*_{\infty} (ℍ)$ is presented.

On a question of T. Sheil-Small regarding valency of harmonic maps

Daoud Bshouty, Abdallah Lyzzaik (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The aim of this work is to answer positively a more general question than the following which is due to T. Sheil-Small: Does the harmonic extension in the open unit disc of a mapping f from the unit circle into itself of the form $f (e^{i t}) = e^{i φ (t)}$ , $0 \leq t \leq 2 π$ where $φ$ is a continuously non-decreasing function that satisfies $φ (2 π) - φ (0) = 2 N π$ , assume every value finitely many times in the disc?

On Polynomially Bounded Harmonic Functions on the $Z^{d}$ Lattice

Piotr Nayar (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove that if $f : Z^{d} \to R$ is harmonic and there exists a polynomial $W : Z^{d} \to R$ such that f + W is nonnegative, then f is a polynomial.

Generalized problem of starlikeness for products of close-to-star functions

Jacek Dziok (2013)

Annales Polonici Mathematici

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We consider functions of the type $F (z) = z \prod_{j = 1}^{n} {[f_{j} (z) / z]}^{a_{j}}$ , where $a_{j}$ are real numbers and $f_{j}$ are $β_{j}$ -strongly close-to-starlike functions of order $α_{j}$ . We look for conditions on the center and radius of the disk (a,r) = z:|z-a| < r, |a| < r ≤ 1 - |a|, ensuring that F((a,r)) is a domain starlike with respect to the origin.

Some properties for $α$ -starlike functions with respect to $k$ -symmetric points of complex order

H. E. Darwish, A. Y. Lashin, S. M. Sowileh (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In the present work, we introduce the subclass $𝒯_{γ, α}^{k} (ϕ)$ , of starlike functions with respect to $k$ -symmetric points of complex order $γ$ ( $γ \neq 0$ ) in the open unit disc . Some interesting subordination criteria, inclusion relations and the integral representation for functions belonging to this class are provided. The results obtained generalize some known results, and some other new results are obtained.

Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales

Adam Osękowski (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let ${(h_{k})}_{k \geq 0}$ be the Haar system on [0,1]. We show that for any vectors $a_{k}$ from a separable Hilbert space and any $ε_{k} \in [- 1, 1]$ , k = 0,1,2,..., we have the sharp inequality $| | \sum_{k = 0}^{n} ε_{k} a_{k} h_{k} {| |}_{W ([0, 1])} \leq 2 | | \sum_{k = 0}^{n} a_{k} h_{k} {| |}_{L^{\infty} ([0, 1])}$ , n = 0,1,2,..., where W([0,1]) is the weak- $L^{\infty}$ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound ${| | Y | |}_{W (Ω)} {\leq 2 | | X | |}_{L^{\infty} (Ω)}$ , where X and Y stand for -valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.

A class of functions containing polyharmonic functions in ℝⁿ

V. Anandam, M. Damlakhi (2003)

Annales Polonici Mathematici

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Some properties of the functions of the form $v (x) = \sum_{i = 0}^{m} {| x |}^{i} h_{i} (x)$ in ℝⁿ, n ≥ 2, where each $h_{i}$ is a harmonic function defined outside a compact set, are obtained using the harmonic measures.

Some characterizations of harmonic Bloch and Besov spaces

Xi Fu, Bowen Lu (2016)

Czechoslovak Mathematical Journal

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The relationship between weighted Lipschitz functions and analytic Bloch spaces has attracted much attention. In this paper, we define harmonic $ω$ - $α$ -Bloch space and characterize it in terms of $ω ((1 - {| x |}^{2})^{β} {(1 - | y |}^{2})^{α - β}) | \frac{f (x) - f (y)}{x - y} |$ and $ω ((1 - {| x |}^{2})^{β} {(1 - | y |}^{2})^{α - β}) | \frac{f (x) - f (y)}{| x | y - x^{'}} |$ where $ω$ is a majorant. Similar results are extended to harmonic little $ω$ - $α$ -Bloch and Besov spaces. Our results are generalizations of the corresponding ones in G. Ren, U. Kähler (2005).

Harmonie reflections

Lieven Vanhecke, Maria-Elena Vazquez-Abal (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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We study local reflections $\phi_{\sigma}$ with respect to a curve $\sigma$ in a Riemannian manifold and prove that $\sigma$ is a geodesic if $\phi_{\sigma}$ is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if $\phi_{\sigma}$ is harmonic for all geodesies $\sigma$ .

A proof of the Livingston conjecture for the fourth and the fifth coefficient of concave univalent functions

Karl-Joachim Wirths (2004)

Annales Polonici Mathematici

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Let D denote the open unit disc and f:D → ℂ̅ be meromorphic and injective in D. We further assume that f has a simple pole at the point p ∈ (0,1) and an expansion $f (z) = z + \sum_{n = 2}^{\infty} a ₙ (f) z ⁿ$ , |z| < p. In particular, we consider f that map D onto a domain whose complement with respect to ℂ̅ is convex. Because of the shape of f(D) these functions will be called concave univalent functions with pole p and the family of these functions is denoted by Co(p). It is proved that for p ∈ (0,1) the domain of variability...

Harmonie reflections

Lieven Vanhecke, Maria-Elena Vazquez-Abal (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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A note on the functions $(S_{n} f) (z)$ defined by Ruscheweyn

S. Owa, C. Y. Shen (1988)

Matematički Vesnik

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Convolution conditions for bounded $α$ -starlike functions of complex order

A. Y. Lashin (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $A$ be the class of analytic functions in the unit disc $U$ of the complex plane $ℂ$ with the normalization $f (0) = f^{^{'}} (0) - 1 = 0$ . We introduce a subclass $S_{M}^{*} (α, b)$ of $A$ , which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class $S_{M}^{*} (n, α, b)$ ( $n \geq 0$ ) related to $S_{M}^{*} (α, b)$ is also considered under the same conditions. Among other things, we find convolution conditions for a function $f \in A$ to belong to the class $S_{M}^{*} (α, b)$ . Several properties of the class $S_{M}^{*} (n, α, b)$ are investigated. ...

Arithmetic theory of harmonic numbers (II)

Zhi-Wei Sun, Li-Lu Zhao (2013)

Colloquium Mathematicae

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For k = 1,2,... let $H_{k}$ denote the harmonic number $\sum_{j = 1}^{k} 1 / j$ . In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have $\sum_{k = 1}^{p - 1} (H_{k}) / (k 2^{k}) \equiv 7 / 24 p B_{p - 3} (m o d p ²)$ , $\sum_{k = 1}^{p - 1} (H_{k, 2}) / (k 2^{k}) \equiv - 3 / 8 B_{p - 3} (m o d p)$ , and $\sum_{k = 1}^{p - 1} (H ²_{k, 2 n}) / (k^{2 n}) \equiv ((\binom{6 n + 1}{2 n - 1}) + n) / (6 n + 1) p B_{p - 1 - 6 n} (m o d p ²)$ for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and $H_{k, m} : = \sum_{j = 1}^{k} 1 / (j^{m})$ .

On the maximum of $a_{4} - 3 a_{2} a_{3} + μ a_{2}$ and some related functionals for bounded real univalent functions

Z. J. Jakubowski, H. Siejka, O. Tammi (1985)

Annales Polonici Mathematici

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On the dimension of $p$ -harmonic measure in space

John L. Lewis, Kaj Nyström, Andrew Vogel (2013)

Journal of the European Mathematical Society

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Let $Ω \subset ℝ^{n}, n \geq 3$ , and let $p, 1 < p < \infty, p \neq 2$ , be given. In this paper we study the dimension of $p$ -harmonic measures that arise from non-negative solutions to the $p$ -Laplace equation, vanishing on a portion of $\partial Ω$ , in the setting of $δ$ -Reifenberg flat domains. We prove, for $p \geq n$ , that there exists $\tilde{δ} = \tilde{δ} (p, n) > 0$ small such that if $Ω$ is a $δ$ -Reifenberg flat domain with $δ < \tilde{δ}$ , then $p$ -harmonic measure is concentrated on a set of $σ$ -finite $H^{n - 1}$ -measure. We prove, for $p \geq n$ , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of $p$ -harmonic...

A Weighted Eigenvalue Problems Driven by both $p (\cdot)$ -Harmonic and $p (\cdot)$ -Biharmonic Operators

Mohamed Laghzal, Abdelouahed El Khalil, Abdelfattah Touzani (2021)

Communications in Mathematics

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The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both $p (\cdot)$ -Harmonic and $p (\cdot)$ -biharmonic operators $\begin{matrix} Δ_{p (x)}^{2} u - Δ_{p (x)} u = λ w (x) {| u |}^{q (x) - 2} u in Ω, \\ u \in W^{2, p (\cdot)} (Ω) \cap W_{0}^{1, p (\cdot)} (Ω), \end{matrix}$ is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces $L^{p (\cdot)} (Ω)$ and $W^{m, p (\cdot)} (Ω)$ .