Displaying similar documents to “Estimates of the real part of f ( z ) z for some classes of univalent functions”

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 , λ ) for 0 z P ( ζ ) d ζ when P ranges over the class ( λ ) = P 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.

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 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.

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 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.

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 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 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 A to belong to the class S M * ( α , b ) . Several properties of the class S M * ( n , α , b ) are investigated. ...

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....

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 ) .

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 + 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.

Injectivity of sections of convex harmonic mappings and convolution theorems

Liulan Li, Saminathan Ponnusamy (2016)

Czechoslovak Mathematical Journal

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We consider the class 0 of sense-preserving harmonic functions f = h + g ¯ defined in the unit disk | z | < 1 and normalized so that h ( 0 ) = 0 = h ' ( 0 ) - 1 and g ( 0 ) = 0 = g ' ( 0 ) , where h and g are analytic in the unit disk. In the first part of the article we present two classes 𝒫 H 0 ( α ) and 𝒢 H 0 ( β ) of functions from 0 and show that if f 𝒫 H 0 ( α ) and F 𝒢 H 0 ( β ) , then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters α and β are satisfied. In the second part we study the harmonic sections...

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

Subclasses of typically real functions determined by some modular inequalities

Leopold Koczan, Katarzyna Trąbka-Więcław (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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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 : | z | < 1 } , normalized by f ( 0 ) = f ' ( 0 ) - 1 = 0 and such that Im z Im f ( z ) 0 for z Δ . Moreover, let us denote: T ( 2 ) : = { f T : f ( z ) = - f ( - z ) for z Δ } and T M , g : = { f T : f M g in Δ } , where M > 1 , g T S and S consists of all analytic functions, normalized and univalent in Δ .We investigate  classes in which the subordination is replaced with the majorization and the function g is typically real but does not necessarily univalent, i.e. classes { f T : f M g in Δ } , where M > 1 , g T , which we denote...

The sizes of the classes of H ( N ) -sets

Václav Vlasák (2014)

Fundamenta Mathematicae

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The class of H ( N ) -sets forms an important subclass of the class of sets of uniqueness for trigonometric series. We investigate the size of this class which is reflected by the family of measures (called polar) annihilating all sets from the class. The main aim of this paper is to answer in the negative a question stated by Lyons, whether the polars of the classes of H ( N ) -sets are the same for all N ∈ ℕ. To prove our result we also present a new description of H ( N ) -sets.

On sentences provable in impredicative extensions of theories

Zygmunt Ratajczyk

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CONTENTS0. Introduction.......................................................................... 51. Preliminaries............................................................................... 72. Basic facts to be used in the sequel....................................... 113. Predicates OD(.,.) and CL(.,.).................................................... 174. Predicate Sels............................................................................. 185. Strong n 1 -collection...........................................................

On C * -spaces

P. Srivastava, K. K. Azad (1981)

Matematički Vesnik

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