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

S. Sivasubramanian; R. Sivakumar; S. Kanas; Seong-A Kim

Displaying similar documents to “Verification of Brannan and Clunie's conjecture for certain subclasses of bi-univalent functions”

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...

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...

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.

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

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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 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 $h$ .

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...

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...

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.

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.

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.

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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Criteria for univalence, starlikeness and convexity

S. Ponnusamy, P. Vasundhra (2005)

Annales Polonici Mathematici

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Let 𝓐 denote the class of all normalized analytic functions f (f(0) = 0 = f'(0)-1) in the open unit disc Δ. For 0 < λ ≤ 1, define 𝓤(λ) = {f ∈ 𝓐 : |(z/f(z))²f'(z) - 1| < λ, z ∈ Δ} and 𝓟(2λ) = f ∈ 𝓐 : |(z/f(z))''| < 2λ, z ∈ Δ.cr Recently, the problem of finding the starlikeness of these classes has been considered by Obradović and Ponnusamy, and later by Obradović et al. In this paper, the authors consider the problem of finding...

Coefficient bounds for some subclasses of p-valently starlike functions

C. Selvaraj, O. S. Babu, G. Murugusundaramoorthy (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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For functions of the form $f (z) = z^{p} + \sum_{n = 1}^{\infty} a_{p + n} z^{p + n}$ we obtain sharp bounds for some coefficients functionals in certain subclasses of starlike functions. Certain applications of our main results are also given. In particular, Fekete-Szego-like inequality for classes of functions defined through extended fractional differintegrals are obtained.

Applications of the Hadamard product in geometric function theory

Zbigniew Jerzy Jakubowski, Piotr Liczberski, Łucja Żywień (1991)

Mathematica Bohemica

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Let $𝒜$ denote the set of functions $F$ holomorphic in the unit disc, normalized clasically: $F (0) = 0, F^{'} (0) = 1$ , whereas $A \subset 𝒜$ is an arbitrarily fixed subset. In this paper various properties of the classes $A_{α}, α \in C {- 1, - \frac{1}{2}, ...}$ , of functions of the form $f = F * k_{α}$ are studied, where $F \in . A$ , $k_{α} (z) = k (z, α) = z + \frac{1}{1 + α} z^{2} + ... + \frac{1}{1 + (n - 1) α} z^{n} + ...$ , and $F * k_{α}$ denotes the Hadamard product of the functions $F$ and $k_{α}$ . Some special cases of the set $A$ were considered by other authors (see, for example, [15],[6],[3]).

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

B. Platynowicz (1980)

Matematički Vesnik

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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.

Univalent harmonic mappings II

Albert E. Livingston (1997)

Annales Polonici Mathematici

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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 $f_{z} ̅ (0) = 0$ .