scholarly journals Estimates of Coefficient Functionals for Functions Convex in the Imaginary-Axis Direction

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1736
Author(s):  
Paweł Zaprawa ◽  
Katarzyna Tra̧bka-Wiȩcław
Keyword(s):  
Imaginary Axis ◽  
Positive Real Part ◽  
Open Unit ◽  
Coefficient Estimates ◽  
Positive Real ◽  
Axis Direction ◽  
Coefficient Problems ◽  
Typically Real ◽  
The Difference ◽  
Typically Real Functions

Let C0(h) be a subclass of analytic and close-to-convex functions defined in the open unit disk by the formula Re{(1−z2)f′(z)}>0. In this paper, some coefficient problems for C0(h) are considered. Some properties and bounds of several coefficient functionals for functions belonging to this class are provided. The main aim of this paper is to find estimates of the difference and of sum of successive coefficients, bounds of the sum of the first n coefficients and bounds of the n-th coefficient. The obtained results are used to determine coefficient estimates for both functions convex in the imaginary-axis direction with real coefficients and typically real functions. Moreover, the sum of the first initial coefficients for functions with a positive real part and with a fixed second coefficient is estimated.

Coefficient Estimates of Some Classes of Analytic Functions

1983 ◽  
Vol 26 (2) ◽  
pp. 202-208
Author(s):  
Nicolas Samaris
Keyword(s):  
Analytic Functions ◽  
Positive Real Part ◽  
Coefficient Estimates ◽  
Positive Real ◽  
Real Functions ◽  
Typically Real ◽  
Typically Real Functions

AbstractWe are concerned with coefficient estimates, and other similar problems, of the typically real functions and of the functions with positive real part. Following the stream of ideas in [1], new results and generalizations of others given in [1], [2] and [3] are obtained.

scholarly journals Generalizing Certain Analytic Functions Correlative to the n-th Coefficient of Certain Class of Bi-Univalent Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Hameed Ur Rehman ◽  
Maslina Darus ◽  
Jamal Salah
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Positive Real Part ◽  
Polynomial Expansion ◽  
Open Unit ◽  
Open Unit Disk ◽  
Positive Real ◽  
Faber Polynomial ◽  
Coefficient Problems

In the present paper, the authors implement the two analytic functions with its positive real part in the open unit disk. New types of polynomials are introduced, and by using these polynomials with the Faber polynomial expansion, a formula is structured to solve certain coefficient problems. This formula is applied to a certain class of bi-univalent functions and solve the n -th term of its coefficient problems. In the last section of the article, several well-known classes are also extended to its n -th term.

scholarly journals Some characteristic properties of analytic functions

2016 ◽  
Vol 24 (1) ◽  
pp. 353-369
Author(s):  
R. K. Raina ◽  
Poonam Sharma ◽  
G. S. Sălăgean
Keyword(s):  
Convex Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Positive Real Part ◽  
Integral Representations ◽  
Open Unit ◽  
Open Unit Disk ◽  
Positive Real ◽  
Bilinear Mapping ◽  
Initial Coefficient

AbstractIn this paper, we consider a class L(λ, μ; ϕ) of analytic functions f defined in the open unit disk U satisfying the subordination condition that,where is the Sălăgean operator and ϕ(z) is a convex function with positive real part in U. We obtain some characteristic properties giving the coefficient inequality, radius and subordination results, and an inclusion result for the above class when the function ϕ(z) is a bilinear mapping in the open unit disk. For these functions f (z) ; sharp bounds for the initial coefficient and for the Fekete-Szegö functional are determined, and also some integral representations are given.

scholarly journals Coefficient estimate of p-valent Bazilevic functions with a bounded positive real part

2015 ◽  
Vol 34 (2) ◽  
pp. 63-73
Author(s):  
O. S. Babu ◽  
C. Selvaraj ◽  
S. Logu ◽  
Gangadharan Murugusundaramoorthy
Keyword(s):  
Unit Disk ◽  
Positive Real Part ◽  
Open Unit ◽  
Open Unit Disk ◽  
Coefficient Estimate ◽  
Positive Real ◽  
Strip Domain ◽  
Bazilevic Functions ◽  
Bounded Positive Real Part

By considering a $p-$valent Bazilevi\v{c} function in the open unit disk$\triangle$ which maps $\triangle$ onto the strip domain $w$ with$p\alpha < \Re\, w < p \beta,$ we estimate bounds of coefficients and solve Fekete-Szeg\"{o} problem forfunctions in this class.\\

scholarly journals Disk-like functions

1970 ◽  
Vol 11 (2) ◽  
pp. 251-256
Author(s):  
Richard J. Libera
Keyword(s):  
Unit Disk ◽  
Geometric Property ◽  
Starlike Functions ◽  
Positive Real Part ◽  
Open Unit ◽  
Open Unit Disk ◽  
Positive Real ◽  
Image Position

The class s of functions f(z) which are regular and univalent in the open unit disk △ = {z: |z| < 1} each normalized by the conditionshas been studied intensively for over fifty years. A large and very successful portion of this work has dealt with subclasses of L characterized by some geometric property of f[Δ], the image of Δ under f(z), which is expressible in analytic terms. The class of starlike functions in L is one of these [3]; f(z) is starlike with respect to the origin if the segment [0,f(z)] is in f[Δ] for every z in Δ and this condition is equivalent to requiring that have a positive real part in Δ.

scholarly journals Subordination of functions in subclass of Bazilevič Functions B 1(α, β)

2021 ◽  
Vol 2106 (1) ◽  
pp. 012026
Author(s):  
Marjono
Keyword(s):  
Unit Disc ◽  
Starlike Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Sharp Estimates ◽  
Strongly Starlike Functions ◽  
Strongly Starlike ◽  
Coefficient Problems ◽  
Bazilevic Functions

Abstract Let f be analytic in the unit disc D = {z : |z| < 1} with f ( z ) = z + ∑ n = 2 ∞ a n z n , and for α ≥ 0 and 0 < β ≤ 1, let B 1(α, ß), denote for the class of Bazilevič functions satisfying the expression | arg z 1 − α f ′ ( z ) f ( z ) 1 − α | < β π 2 . We give sharp estimates for various coefficient problems for functions in B 1(α, β), which unify and extend well-known results for starlike functions, strongly starlike functions and functions whose derivative has positive real part in domain D.

scholarly journals On the Fekete–Szegö Type Functionals for Close-to-Convex Functions

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1497 ◽  
Author(s):  
Katarzyna Tra̧bka-Wiȩcław ◽  
Paweł Zaprawa ◽  
Magdalena Gregorczyk ◽  
Andrzej Rysak
Keyword(s):  
Analytic Function ◽  
Real Number ◽  
Convex Functions ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Hankel Determinants ◽  
Coefficient Problems ◽  
Theoretical Results

In this paper, we consider two functionals of the Fekete–Szegö type Θ f ( μ ) = a 4 − μ a 2 a 3 and Φ f ( μ ) = a 2 a 4 − μ a 3 2 for a real number μ and for an analytic function f ( z ) = z + a 2 z 2 + a 3 z 3 + … , | z | < 1 . This type of research was initiated by Hayami and Owa in 2010. They obtained results for functions satisfying one of the conditions Re f ( z ) / z > α or Re f ′ ( z ) > α , α ∈ [ 0 , 1 ) . Similar estimates were also derived for univalent starlike functions and for univalent convex functions. We discuss Θ f ( μ ) and Φ f ( μ ) for close-to-convex functions such that f ′ ( z ) = h ( z ) / ( 1 − z ) 2 , where h is an analytic function with a positive real part. Many coefficient problems, among others estimating of Θ f ( μ ) , Φ f ( μ ) or the Hankel determinants for close-to-convex functions or univalent functions, are not solved yet. Our results broaden the scope of theoretical results connected with these functionals defined for different subclasses of analytic univalent functions.

scholarly journals Radius of starlikeness of certain analytic functions

2021 ◽  
Vol 71 (1) ◽  
pp. 83-104
Author(s):  
Asha Sebastian ◽  
V. Ravichandran
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Starlike Functions ◽  
Positive Real Part ◽  
Open Unit ◽  
Open Unit Disk ◽  
Sine Function ◽  
Positive Real ◽  
Radius Of Starlikeness ◽  
Lemniscate Of Bernoulli

Abstract This paper studies analytic functions f defined on the open unit disk of the complex plane for which f/g and (1 + z)g/z are both functions with positive real part for some analytic function g. We determine radius constants of these functions to belong to classes of strong starlike functions, starlike functions of order α, parabolic starlike functions, as well as to the classes of starlike functions associated with lemniscate of Bernoulli, cardioid, lune, reverse lemniscate, sine function, exponential function and a particular rational function. The results obtained are sharp.

scholarly journals Starlikeness of certain non-univalent functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Adam Lecko ◽  
V. Ravichandran ◽  
Asha Sebastian
Keyword(s):  
Analytic Functions ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Positive Real Part ◽  
The Other ◽  
Open Unit ◽  
Sigmoid Function ◽  
Open Unit Disk ◽  
Positive Real ◽  
Lemniscate Of Bernoulli

AbstractWe consider three classes of functions defined using the class $${\mathcal {P}}$$ P of all analytic functions $$p(z)=1+cz+\cdots $$ p ( z ) = 1 + c z + ⋯ on the open unit disk having positive real part and study several radius problems for these classes. The first class consists of all normalized analytic functions f with $$f/g\in {\mathcal {P}}$$ f / g ∈ P and $$g/(zp)\in {\mathcal {P}}$$ g / ( z p ) ∈ P for some normalized analytic function g and $$p\in {\mathcal {P}}$$ p ∈ P . The second class is defined by replacing the condition $$f/g\in {\mathcal {P}}$$ f / g ∈ P by $$|(f/g)-1|<1$$ | ( f / g ) - 1 | < 1 while the other class consists of normalized analytic functions f with $$f/(zp)\in {\mathcal {P}}$$ f / ( z p ) ∈ P for some $$p\in {\mathcal {P}}$$ p ∈ P . We have determined radii so that the functions in these classes to belong to various subclasses of starlike functions. These subclasses includes the classes of starlike functions of order $$\alpha $$ α , parabolic starlike functions, as well as the classes of starlike functions associated with lemniscate of Bernoulli, reverse lemniscate, sine function, a rational function, cardioid, lune, nephroid and modified sigmoid function.

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