Initial coefficients for generalized subclasses of bi-univalent functions defined with subordination

Gagandeep Singh; Gurcharanjit Singh; Gurmeet Singh

Displaying similar documents to “Initial coefficients for generalized subclasses of bi-univalent functions defined with subordination”

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...

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 $h$ .

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...

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...

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...

Applications of the theory of differential subordination for functions with fixed initial coefficient to univalent functions

Sumit Nagpal, V. Ravichandran (2012)

Annales Polonici Mathematici

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By using the theory of first-order differential subordination for functions with fixed initial coefficient, several well-known results for subclasses of univalent functions are improved by restricting the functions to have fixed second coefficient. The influence of the second coefficient of univalent functions becomes evident in the results obtained.