Applications of the Hadamard product in geometric function theory

Zbigniew Jerzy Jakubowski; Piotr Liczberski; Łucja Żywień

Displaying similar documents to “Applications of the Hadamard product in geometric function theory”

On some extremal problems in classes of holomorphic functions $S_{p}^{} (ρ), S_{p}^{\circ} (ρ), Σ_{p}^{} (ρ)$

B. Platynowicz (1980)

Matematički Vesnik

Similarity:

Verification of Brannan and Clunie's conjecture for certain subclasses of bi-univalent functions

S. Sivasubramanian, R. Sivakumar, S. Kanas, Seong-A Kim (2015)

Annales Polonici Mathematici

Similarity:

Let σ denote the class of bi-univalent functions f, that is, both f(z) = z + a₂z² + ⋯ and its inverse $f^{- 1}$ are analytic and univalent on the unit disk. We consider the classes of strongly bi-close-to-convex functions of order α and of bi-close-to-convex functions of order β, which turn out to be subclasses of σ. We obtain upper bounds for |a₂| and |a₃| for those classes. Moreover, we verify Brannan and Clunie’s conjecture |a₂| ≤ √2 for some of our classes. In addition, we obtain the Fekete-Szegö...

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

Adam Lecko (2002)

Annales Polonici Mathematici

Similarity:

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.

Coefficient inequalities for concave and meromorphically starlike univalent functions

B. Bhowmik, S. Ponnusamy (2008)

Annales Polonici Mathematici

Similarity:

Let denote the open unit disk and f: → ℂ̅ be meromorphic and univalent in with a simple pole at p ∈ (0,1) and satisfying the standard normalization f(0) = f’(0)-1 = 0. Also, assume that f has the expansion $f (z) = \sum_{n = - 1}^{\infty} a ₙ (z - p) ⁿ$ , |z-p| < 1-p, and maps onto a domain whose complement with respect to ℂ̅ is a convex set (starlike set with respect to a point w₀ ∈ ℂ, w₀ ≠ 0 resp.). We call such functions concave (meromorphically starlike resp.) univalent functions and denote this class by $C o (p) (Σ^{s} (p, w ₀)$ resp.). We prove...

Certain subclasses of starlike functions of complex order involving the Hurwitz-Lerch Zeta function

G. Murugusundaramoorthy, K. Uma (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

Making use of the Hurwitz-Lerch Zeta function, we define a new subclass of uniformly convex functions and a corresponding subclass of starlike functions with negative coefficients of complex order denoted by $T S_{b}^{μ} (α, β, γ)$ and obtain coefficient estimates, extreme points, the radii of close to convexity, starlikeness and convexity and neighbourhood results for the class $T S_{b}^{μ} (α, β, γ)$ . In particular, we obtain integral means inequalities for the function $f (z)$ belongs to the class $T S_{b}^{μ} (α, β, γ)$ in the unit disc.

Sufficient conditions for starlike and convex functions

S. Ponnusamy, P. Vasundhra (2007)

Annales Polonici Mathematici

Similarity:

For n ≥ 1, let denote the class of all analytic functions f in the unit disk Δ of the form $f (z) = z + \sum_{k = 2}^{\infty} a_{k} z^{k}$ . For Re α < 2 and γ > 0 given, let (γ,α) denote the class of all functions f ∈ satisfying the condition |f’(z) - α f(z)/z + α - 1| ≤ γ, z ∈ Δ. We find sufficient conditions for functions in (γ,α) to be starlike of order β. A generalization of this result along with some convolution results is also obtained.

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

M. Obradović (1984)

Matematički Vesnik

Similarity:

Fekete–Szegö Problem for a New Class of Analytic Functions Defined by Using a Generalized Differential Operator

M. K. Aouf, R. M. El-Ashwah, A. A. M. Hassan, A. H. Hassan (2013)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Similarity:

In this paper, we obtain Fekete–Szegö inequalities for a generalized class of analytic functions $f (z) \in 𝒜$ for which $1 + \frac{1}{b} (\frac{z {(D_{α, β, λ, δ}^{n} f (z))}^{'}}{D_{α, β, λ, δ}^{n} f (z)} - 1)$ ( $α, β, λ, δ \geq 0$ ; $β > α$ ; $λ > δ$ ; $b \in ℂ^{*}$ ; $n \in ℕ_{0}$ ; $z \in U$ ) lies in a region starlike with respect to $1$ and is symmetric with respect to the real axis.

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

Karl-Joachim Wirths (2004)

Annales Polonici Mathematici

Similarity:

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...

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

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

Matematički Vesnik

Similarity:

Univalent harmonic mappings II

Albert E. Livingston (1997)

Annales Polonici Mathematici

Similarity:

Let a < 0 < b and Ω(a,b) = ℂ - ((-∞, a] ∪ [b,+∞)) and U= z: |z| < 1. We consider the class $S_{H} (U, Ω (a, b))$ of functions f which are univalent, harmonic and sense-preserving with f(U) = Ω and satisfying f(0) = 0, $f_{z} (0) > 0$ and $f_{z} ̅ (0) = 0$ .

Region of variability for functions with positive real part

Saminathan Ponnusamy, Allu Vasudevarao (2010)

Annales Polonici Mathematici

Similarity:

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.

Spirallike mappings and univalent subordination chains in $ℂ^{n}$

Ian Graham, Hidetaka Hamada, Gabriela Kohr, Mirela Kohr (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

In this paper we consider non-normalized univalent subordination chains and the connection with the Loewner differential equation on the unit ball in $ℂ^{n}$ . To this end, we study the most general form of the initial value problem for the transition mapping, and prove the existence and uniqueness of solutions. Also we introduce the notion of generalized spirallikeness with respect to a measurable matrix-valued mapping, and investigate this notion from the point of view of non-normalized univalent...

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

Jacek Dziok (2013)

Annales Polonici Mathematici

Similarity:

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

Similarity:

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.

Sets in $ℂ^{N}$ with vanishing global extremal function and polynomial approximation

Józef Siciak (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

Similarity:

Let $Γ$ be a non-pluripolar set in $ℂ^{N}$ . Let $f$ be a function holomorphic in a connected open neighborhood $G$ of $Γ$ . Let ${P_{n}}$ be a sequence of polynomials with $deg P_{n} \leq d_{n} (d_{n} < d_{n + 1})$ such that $\underset{n \to \infty}{lim sup} {| f (z) - P_{n} (z) |}^{1 / d_{n}} < 1, z \in Γ .$ We show that if $\underset{n \to \infty}{lim sup} {| P_{n} (z) |}^{1 / d_{n}} \leq 1, z \in E,$ where $E$ is a set in $ℂ^{N}$ such that the global extremal function $V_{E} \equiv 0$ in $ℂ^{N}$ , then the maximal domain of existence $G_{f}$ of $f$ is one-sheeted, and $\underset{n \to \infty}{lim sup} {∥ f - P_{n} ∥}_{K}^{\frac{1}{d_{n}}} < 1$ for every compact set $K \subset G_{f}$ . If, moreover, the sequence ${d_{n + 1} / d_{n}}$ is bounded then $G_{f} = ℂ^{N}$ . If $E$ is a closed...