A note on the functions defined by Ruscheweyn
S. Owa, C. Y. Shen (1988)
Matematički Vesnik
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S. Owa, C. Y. Shen (1988)
Matematički Vesnik
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M. Obradović (1984)
Matematički Vesnik
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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 denote the class of all analytic functions P in the unit disk with P(0) = 1 and in . For any fixed z₀ ∈ and λ ∈ ̅, we shall determine the region of variability for when P ranges over the class 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.
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 and have also been estimated.
Adam Lecko (2002)
Annales Polonici Mathematici
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The purpose of this paper is to study the class 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 is presented.
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 , of starlike functions with respect to -symmetric points of complex order () 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.
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 -pseudo-starlike functions with Sakaguchi-type functions. We determine upper bounds for the Taylor–Maclaurin coefficients and for functions belonging to these classes. Further we point out certain special cases for our results.
Jacek Dziok (2013)
Annales Polonici Mathematici
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We consider functions of the type , where are real numbers and are -strongly close-to-starlike functions of order . 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.
Lucjan Siewierski
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CONTENTSIntroduction...............................................................................................................................................................................5Definitions and notation.........................................................................................................................................................7The main result........................................................................................................................................................................91....
A. Y. Lashin (2017)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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Let be the class of analytic functions in the unit disc of the complex plane with the normalization . We introduce a subclass of , which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class () related to is also considered under the same conditions. Among other things, we find convolution conditions for a function to belong to the class . Several properties of the class are investigated. ...
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 , |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...
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 and obtain coefficient estimates, extreme points, the radii of close to convexity, starlikeness and convexity and neighbourhood results for the class . In particular, we obtain integral means inequalities for the function belongs to the class in the unit disc.
Liulan Li, Saminathan Ponnusamy (2016)
Czechoslovak Mathematical Journal
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We consider the class of sense-preserving harmonic functions defined in the unit disk and normalized so that and , where and are analytic in the unit disk. In the first part of the article we present two classes and of functions from and show that if and , then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters and are satisfied. In the second part we study the harmonic sections...
Mamoru Nunokawa, Emel Yavuz Duman, Shigeyoshi Owa (2013)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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Let denote the class of analytic functions of the form in the open unit disc . Applying the result by S. S. Miller and P. T. Mocanu (J. Math. Anal. Appl. 65 (1978), 289-305), some interesting properties for 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.
Wafaa Y. Kota, Rabha Mohamed El-Ashwah (2023)
Mathematica Bohemica
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Using the operator , we introduce the subclasses and 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.
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 and from the class and also we estimate the first two coefficients for functions obtained by this operator applied on the class .
B. Platynowicz (1980)
Matematički Vesnik
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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 and to be univalent in the open unit disc , where the functions belong to the classes and . The order of convexity for the operators and is also determined. Furthermore, and for , we obtain sufficient conditions for the operators and to be in the class . Several corollaries and consequences of the main results are also considered.
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 of analytic functions in the unit disk. We also investigate several majorization properties for functions belonging to a subclass of and related function classes. Relevant connections of the main results obtained here with those given by earlier workers on the subject are pointed out.
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 , and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space can be uniformly approximated by smooth Lipschitz functions so that the cones generated by the ranges of its derivatives have empty interior. This implies that there are smooth...
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 . For λ > 0, suppose that denotes any one of the following classes of functions: , , . The main purpose of this paper is to find conditions on λ and γ so that each f ∈ is in or , γ ∈ [0,1/2]. Here and 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...
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 denote the class of analytic functions with the normalization in the open unit disc . Set and define in terms of the Hadamard product In this paper, we introduce several subclasses of analytic functions defined by means of the operator , given by Inclusion properties of these classes and the classes involving the generalized Libera integral operator are also considered.
Z. J. Jakubowski, H. Siejka, O. Tammi (1985)
Annales Polonici Mathematici
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Ding-Gong Yang, Jin-Lin Liu (2014)
Annales Polonici Mathematici
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The authors introduce two new subclasses and of meromorphically multivalent functions. Distortion bounds and convolution properties for , and their subclasses with positive coefficients are obtained. Some inclusion relations for these function classes are also given.
Leopold Koczan, Katarzyna Trąbka-Więcław (2010)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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Let be the family of all typically real functions, i.e. functions that are analytic in the unit disk , normalized by and such that Im Im for . Moreover, let us denote: and , where , and consists of all analytic functions, normalized and univalent in .We investigate classes in which the subordination is replaced with the majorization and the function is typically real but does not necessarily univalent, i.e. classes , where , , which we denote...
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 of functions f which are univalent, harmonic and sense-preserving with f(U) = Ω and satisfying f(0) = 0, and .