On products of starlike functions. I

Georgi Dimkov

Displaying similar documents to “On products of starlike functions. I”

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.

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.

On some simple sufficient conditions for univalence

Nikola Tuneski (2001)

Mathematica Bohemica

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In this paper some simple conditions on $f^{'} (z)$ and $f^{''} (z)$ which lead to some subclasses of univalent functions will be considered.

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...

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 $h$ .

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 $T_{g}$ .

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...

Subordination criteria for starlikeness and convexity.

Aghalary, Rasoul, Jahangiri, Jay M. (2003)

International Journal of Mathematics and Mathematical Sciences

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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.