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

Adam Lecko

Displaying similar documents to “On the class of functions defined in a halfplane and starlike with respect to the boundary point”

Certain subclass of alpha-convex bi-univalent functions defined using $q$ -derivative operator

Gagandeep Singh, Gurcharanjit Singh (2025)

Archivum Mathematicum

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The present investigation deals with a new subclass of alpha-convex bi-univalent functions in the unit disc $E = \}z : ∣ z ∣ < 1\{$ defined with $q$ -derivative operator. Bounds for the first two coefficients and Fekete-Szegö inequality are established for this class. Many known results follow as consequences of the results derived here.

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.

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.

Sakaguchi type functions defined by balancing polynomials

Gunasekar Saravanan, Sudharsanan Baskaran, Balasubramaniam Vanithakumari, Serap Bulut (2025)

Mathematica Bohemica

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The class of Sakaguchi type functions defined by balancing polynomials has been introduced as a novel subclass of bi-univalent functions. The bounds for the Fekete-Szegö inequality and the initial coefficients $| a_{2} |$ and $| a_{3} |$ have also been estimated.

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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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

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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.

Initial Maclaurin coefficient estimates for $λ$ -pseudo-starlike bi-univalent functions associated with Sakaguchi-type functions

Abbas Kareem Wanas, Basem Aref Frasin (2022)

Mathematica Bohemica

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We introduce and study two certain classes of holomorphic and bi-univalent functions associating $λ$ -pseudo-starlike functions with Sakaguchi-type functions. We determine upper bounds for the Taylor–Maclaurin coefficients $| a_{2} |$ and $| a_{3} |$ for functions belonging to these classes. Further we point out certain special cases for our results.

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

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

Matematički Vesnik

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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.

Properties of functions concerned with Caratheodory functions

Mamoru Nunokawa, Emel Yavuz Duman, Shigeyoshi Owa (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $𝒫_{n}$ denote the class of analytic functions $p (z)$ of the form $p (z) = 1 + c_{n} z^{n} + c_{n + 1} z^{n + 1} + \dots$ in the open unit disc $𝕌$ . Applying the result by S. S. Miller and P. T. Mocanu (J. Math. Anal. Appl. 65 (1978), 289-305), some interesting properties for $p (z)$ concerned with Caratheodory functions are discussed. Further, some corollaries of the results concerned with the result due to M. Obradovic and S. Owa (Math. Nachr. 140 (1989), 97-102) are shown.

Some applications of subordination theorems associated with fractional $q$ -calculus operator

Wafaa Y. Kota, Rabha Mohamed El-Ashwah (2023)

Mathematica Bohemica

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Using the operator $𝔇_{q, ϱ}^{m} (λ, l)$ , we introduce the subclasses $𝔜_{q, ϱ}^{* m} (l, λ, γ)$ and $𝔎_{q, ϱ}^{* m} (l, λ, γ)$ of normalized analytic functions. Among the results investigated for each of these function classes, we derive some subordination results involving the Hadamard product of the associated functions. The interesting consequences of some of these subordination results are also discussed. Also, we derive integral means results for these classes.

Convolution conditions for bounded $α$ -starlike functions of complex order

A. Y. Lashin (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $A$ be the class of analytic functions in the unit disc $U$ of the complex plane $ℂ$ with the normalization $f (0) = f^{^{'}} (0) - 1 = 0$ . We introduce a subclass $S_{M}^{*} (α, b)$ of $A$ , which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class $S_{M}^{*} (n, α, b)$ ( $n \geq 0$ ) related to $S_{M}^{*} (α, b)$ is also considered under the same conditions. Among other things, we find convolution conditions for a function $f \in A$ to belong to the class $S_{M}^{*} (α, b)$ . Several properties of the class $S_{M}^{*} (n, α, b)$ are investigated. ...

On certain subclasses of analytic functions associated with the Carlson–Shaffer operator

Jagannath Patel, Ashok Kumar Sahoo (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The object of the present paper is to solve Fekete-Szego problem and determine the sharp upper bound to the second Hankel determinant for a certain class $R^{λ} (a, c, A, B)$ of analytic functions in the unit disk. We also investigate several majorization properties for functions belonging to a subclass ${\tilde{R}}^{λ} (a, c, A, B)$ of $R^{λ} (a, c, A, B)$ and related function classes. Relevant connections of the main results obtained here with those given by earlier workers on the subject are pointed out.

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.

Sharp estimation of the coefficients of bounded univalent functions close to identity

Lucjan Siewierski

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CONTENTSIntroduction...............................................................................................................................................................................5Definitions and notation.........................................................................................................................................................7The main result........................................................................................................................................................................91....

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.

Region of variability for spiral-like functions with respect to a boundary point

S. Ponnusamy, A. Vasudevarao, M. Vuorinen (2009)

Colloquium Mathematicae

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For μ ∈ ℂ such that Re μ > 0 let $ℱ_{μ}$ denote the class of all non-vanishing analytic functions f in the unit disk with f(0) = 1 and $R e (2 π / μ z f^{'} (z) / f (z) + (1 + z) / (1 - z)) > 0$ in . For any fixed z₀ in the unit disk, a ∈ ℂ with |a| ≤ 1 and λ ∈ ̅, we shall determine the region of variability V(z₀,λ) for log f(z₀) when f ranges over the class $ℱ_{μ} (λ) = f \in ℱ_{μ} : f^{'} (0) = (μ / π) (λ - 1) a n d f^{''} (0) = (μ / π) (a (1 - | λ | ²) + (μ / π) (λ - 1) ² - (1 - λ ²))$ . In the final section we graphically illustrate the region of variability for several sets of parameters.

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...

Some properties of certain subclasses of bounded Mocanu variation with respect to $2 k$ -symmetric conjugate points

Rasoul Aghalary, Jafar Kazemzadeh (2019)

Mathematica Bohemica

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We introduce subclasses of analytic functions of bounded radius rotation, bounded boundary rotation and bounded Mocanu variation with respect to $2 k$ -symmetric conjugate points and study some of its basic properties.

Inclusion properties of certain subclasses of analytic functions defined by generalized Salagean operator

M. K. Aouf, A. Shamandy, A. O. Mostafa, S. M. Madian (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $A$ denote the class of analytic functions with the normalization $f (0) = f^{'} (0) - 1 = 0$ in the open unit disc $U = {z : |z| < 1}$ . Set $f_{λ}^{n} (z) = z + \sum_{k = 2}^{\infty} {[1 + λ (k - 1)]}^{n} z^{k} (n \in N_{0}; λ \geq 0; z \in U),$ and define $f_{λ, μ}^{n}$ in terms of the Hadamard product $f_{λ}^{n} (z) * f_{λ, μ}^{n} = \frac{z}{{(1 - z)}^{μ}} (μ > 0; z \in U) .$ In this paper, we introduce several subclasses of analytic functions defined by means of the operator $I_{λ, μ}^{n} : A ⟶ A$ , given by $I_{λ, μ}^{n} f (z) = f_{λ, μ}^{n} (z) * f (z) (f \in A; n \in N_{0;} λ \geq 0; μ > 0) .$ Inclusion properties of these classes and the classes involving the generalized Libera integral operator are also considered.

An integral operator on the classes $𝒮^{*} (α)$ and $𝒞𝒱ℋ (β)$

Nicoleta Ularu, Nicoleta Breaz (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The purpose of this paper is to study some properties related to convexity order and coefficients estimation for a general integral operator. We find the convexity order for this operator, using the analytic functions from the class of starlike functions of order $α$ and from the class $𝒞𝒱ℋ (β)$ and also we estimate the first two coefficients for functions obtained by this operator applied on the class $𝒞𝒱ℋ (β)$ .

On the maximum of $a_{4} - 3 a_{2} a_{3} + μ a_{2}$ and some related functionals for bounded real univalent functions

Z. J. Jakubowski, H. Siejka, O. Tammi (1985)

Annales Polonici Mathematici

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Radii of p-valence of certain analytic functions of order $α$ with negative coefficients

M. K. Aouf (1989)

Matematički Vesnik

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Convolution theorems for starlike and convex functions in the unit disc

M. Anbudurai, R. Parvatham, S. Ponnusamy, V. Singh (2004)

Annales Polonici Mathematici

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Let A denote the space of all analytic functions in the unit disc Δ with the normalization f(0) = f’(0) − 1 = 0. For β < 1, let $P ⁰_{β} = f \in A : R e f^{'} (z) > β, z \in Δ$ . For λ > 0, suppose that denotes any one of the following classes of functions: $M_{1, λ}^{(1)} = f \in : R e z {(z f^{'} (z))}^{''} > - λ, z \in Δ$ , $M_{1, λ}^{(2)} = f \in : R e z {(z ² f^{''} (z))}^{''} > - λ, z \in Δ$ , $M_{1, λ}^{(3)} = f \in : R e 1 / 2 {(z {(z ² f^{'} (z))}^{''})}^{'} - 1 > - λ, z \in Δ$ . The main purpose of this paper is to find conditions on λ and γ so that each f ∈ is in $_{γ}$ or $_{γ}$ , γ ∈ [0,1/2]. Here $_{γ}$ and $_{γ}$ respectively denote the class of all starlike functions of order γ and the class of all convex functions of order γ. As a consequence, we obtain...

On certain general integral operators of analytic functions

B. A. Frasin (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper, we obtain new sufficient conditions for the operators $F_{α_{1}, α_{2}, . . ., α_{n}, β} (z)$ and $G_{α_{1}, α_{2}, . . ., α_{n}, β} (z)$ to be univalent in the open unit disc $𝒰$ , where the functions $f_{1}, f_{2}, . . ., f_{n}$ belong to the classes $S^{*} (a, b)$ and $𝒦 (a, b)$ . The order of convexity for the operators $F_{α_{1}, α_{2}, . . ., α_{n}, β} (z)$ and $G_{α_{1}, α_{2}, . . ., α_{n}, β} (z)$ is also determined. Furthermore, and for $β = 1$ , we obtain sufficient conditions for the operators $F_{n} (z)$ and $G_{n} (z)$ to be in the class $𝒦 (a, b)$ . Several corollaries and consequences of the main results are also considered.

On the size of the sets of gradients of bump functions and starlike bodies on the Hilbert space

Daniel Azagra, Mar Jiménez-Sevilla (2002)

Bulletin de la Société Mathématique de France

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We study the size of the sets of gradients of bump functions on the Hilbert space $ℓ_{2}$ , and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in $ℓ_{2}$ can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space $ℓ_{2}$ can be uniformly approximated by $C^{1}$ smooth Lipschitz functions $ψ$ so that the cones generated by the ranges of its derivatives $ψ^{'} (ℓ_{2})$ have empty interior. This implies that there are $C^{1}$ smooth...