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

Displaying similar documents to “Fekete–Szegö Problem for a New Class of Analytic Functions Defined by Using a Generalized Differential Operator”

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.

Sufficient conditions for starlike and convex functions

S. Ponnusamy, P. Vasundhra (2007)

Annales Polonici Mathematici

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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.

A note on Briot-Bouquet-Bernoulli differential subordination

Stanisława Kanas, Joanna Kowalczyk (2005)

Commentationes Mathematicae Universitatis Carolinae

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Let $p, q$ be analytic functions in the unit disk $𝒰$ . For $α \in [0, 1)$ the authors consider the differential subordination and the differential equation of the Briot-Bouquet type: $p^{1 - α} (z) + \frac{z p^{'} (z)}{δ p^{α} (z) + λ p (z)} ≺ h (z), z \in 𝒰,$ $q^{1 - α} (z) + \frac{n z q^{'} (z)}{δ q^{α} (z) + λ q (z)} = h (z), z \in 𝒰,$ with $p (0) = q (0) = h (0) = 1$ . The aim of the paper is to find the dominant and the best dominant of the above subordination. In addition, the authors give some particular cases of the main result obtained for appropriate choices of functions $h$ .

On typically real functions which are generated by a fixed typically real function

Magdalena Sobczak-Kneć, Katarzyna Trąbka-Więcław (2011)

Czechoslovak Mathematical Journal

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Let $T$ be the family of all typically real functions, i.e. functions that are analytic in the unit disk $Δ : = {z \in ℂ : | z | < 1}$ , normalized by $f (0) = f^{'} (0) - 1 = 0$ and such that $Im z Im f (z) \geq 0$ for $z \in Δ$ . In this paper we discuss the class $T_{g}$ defined as $T_{g} : = {\sqrt{f (z) g (z)} : f \in T}, g \in T .$ We determine the sets $⋃_{g \in T} T_{g}$ and $⋂_{g \in T} T_{g}$ . Moreover, for a fixed $g$ , we determine the superdomain of local univalence of $T_{g}$ , the radii of local univalence, of starlikeness and of univalence of $T_{g}$ .

Coefficient inequalities for concave and meromorphically starlike univalent functions

B. Bhowmik, S. Ponnusamy (2008)

Annales Polonici Mathematici

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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...

On products of starlike functions. I

Georgi Dimkov (1991)

Annales Polonici Mathematici

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We deal with functions given by the formula $F (z) = z G^{'} (z) = z \prod_{j = 1}^{n} {(f_{j} (z) / z)}^{a_{j}}$ where $f_{j} (z)$ are starlike of order $α_{j}$ and $a_{j}$ are complex constants. In particular, radii of starlikeness and convexity as well as orders of starlikeness and convexity are found.

Univalence, strong starlikeness and integral transforms

M. Obradović, S. Ponnusamy, P. Vasundhra (2005)

Annales Polonici Mathematici

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Let represent the class of all normalized analytic functions f in the unit disc Δ. In the present work, we first obtain a necessary condition for convex functions in Δ. Conditions are established for a certain combination of functions to be starlike or convex in Δ. Also, using the Hadamard product as a tool, we obtain sufficient conditions for functions to be in the class of functions whose real part is positive. Moreover, we derive conditions on f and μ so that the non-linear integral...

Strongly gamma-starlike functions of order alpha

Mamoru Nunokawa, Janusz Sokół (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this work we consider the class of analytic functions $𝒢 (α, γ)$ , which is a subset of gamma-starlike functions introduced by Lewandowski, Miller and Złotkiewicz in Gamma starlike functions, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 28 (1974), 53–58. We discuss the order of strongly starlikeness and the order of strongly convexity in this subclass.

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

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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...

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

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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...

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

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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.

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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Inclusion and neighborhood properties of certain subclasses of p-valent functions of complex order defined by convolution

R. M. El-Ashwah, M. K. Aouf, S. M. El-Deeb (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we introduce and investigate three new subclasses of $p$ -valent analytic functions by using the linear operator $D_{λ, p}^{m} (f * g) (z)$ . The various results obtained here for each of these function classes include coefficient bounds, distortion inequalities and associated inclusion relations for $(n, θ)$ -neighborhoods of subclasses of analytic and multivalent functions with negative coefficients, which are defined by means of a non-homogenous differential equation.

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.

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.

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

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

Matematički Vesnik

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