scholarly journals Subclasses of close-to-convex functions

1983 ◽  
Vol 6 (3) ◽  
pp. 449-458 ◽  
Author(s):  
E. M. Silvia
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
Integral Operators ◽  
Coefficient Bounds ◽  
Class Of Functions

Let𝒦[C,D],−1≤D<C≤1, denote the class of functionsg(z),g(0)=g′(0)−1=0, analytic in the unit diskU={z:|z|<1}such that1+(zg″(z)/g′(z))is subordinate to(1+Cz)/(1+Dz),z ϵ U. We investigate the subclasses of close-to-convex functionsf(z),f(0)=f′(0)−1=0, for which there existsg ϵ 𝒦[C,D]such thatf′/g′is subordinate to(1+Az)/(1+Bz),−1≤B<A≤1. Distortion and rotation theorems and coefficient bounds are obtained. It is also shown that these classes are preserved under certain integral operators.

scholarly journals Some classes of alpha-quasi-convex functions

1988 ◽  
Vol 11 (3) ◽  
pp. 497-501 ◽  
Author(s):  
Khalida Inayat Noor
Keyword(s):  
Integral Representation ◽  
Unit Disk ◽  
Convex Functions ◽  
Integral Operators ◽  
Coefficient Bounds ◽  
Class Of Functions

LetC[C,D],−1≤D<C≤1denote the class of functionsg,g(0)=0,g′(0)=1, analytic in the unit diskEsuch that(zg′(z))′g′(z)is subordinate to1+CZ1+DZ,z∈E. We investigate some classes of Alpha-Quasi-Convex Functionsf, withf(0)=f′(0)−1=0for which there exists ag∈C[C,D]such that(1−α)f′(z)g′(z)+α(zf′(z))′g′(z)is subordinate to1+AZ1+BZ′,−1≤B<A≤1. Integral representation, coefficient bounds are obtained. It is shown that some of these classes are preserved under certain integral operators.

scholarly journals Logarithmic Coefficient Bounds and Coefficient Conjectures for Classes Associated with Convex Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Davood Alimohammadi ◽  
Ebrahim Analouei Adegani ◽  
Teodor Bulboacă ◽  
Nak Eun Cho
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
Univalent Functions ◽  
Current Paper ◽  
Open Unit ◽  
Open Unit Disk ◽  
Coefficient Bounds ◽  
Logarithmic Coefficients ◽  
Class Of Functions ◽  
Logarithmic Coefficient

It is well-known that the logarithmic coefficients play an important role in the development of the theory of univalent functions. If S denotes the class of functions f z = z + ∑ n = 2 ∞ a n z n analytic and univalent in the open unit disk U , then the logarithmic coefficients γ n f of the function f ∈ S are defined by log f z / z = 2 ∑ n = 1 ∞ γ n f z n . In the current paper, the bounds for the logarithmic coefficients γ n for some well-known classes like C 1 + α z for α ∈ 0 , 1 and C V hpl 1 / 2 were estimated. Further, conjectures for the logarithmic coefficients γ n for functions f belonging to these classes are stated. For example, it is forecasted that if the function f ∈ C 1 + α z , then the logarithmic coefficients of f satisfy the inequalities γ n ≤ α / 2 n n + 1 , n ∈ ℕ . Equality is attained for the function L α , n , that is, log L α , n z / z = 2 ∑ n = 1 ∞ γ n L α , n z n = α / n n + 1 z n + ⋯ , z ∈ U .

scholarly journals Some new generalizations of Ostrowski type inequalities for s-convex functions via fractional integral operators

Filomat ◽  
2018 ◽  
Vol 32 (16) ◽  
pp. 5595-5609
Author(s):  
Erhan Set
Keyword(s):  
Integral Operator ◽  
Convex Functions ◽  
Fractional Integral ◽  
Present Note ◽  
Integral Operators ◽  
Fractional Integral Operator ◽  
Integral Inequalities ◽  
Fractional Integral Operators ◽  
Special Cases ◽  
Class Of Functions

Remarkably a lot of Ostrowski type inequalities involving various fractional integral operators have been investigated by many authors. Recently, Raina [34] introduced a new generalization of the Riemann-Liouville fractional integral operator involving a class of functions defined formally by F? ?,?(x)=??,k=0 ?(k)/?(?k + ?)xk. Using this fractional integral operator, in the present note, we establish some new fractional integral inequalities of Ostrowski type whose special cases are shown to yield corresponding inequalities associated with Riemann-Liouville fractional integral operators.

scholarly journals Bounded functions starlike with respect to symmetrical points

1996 ◽  
Vol 19 (3) ◽  
pp. 615-623
Author(s):  
Fatima M. Al-Oboudi
Keyword(s):  
Unit Disk ◽  
Coefficient Bounds ◽  
Bounded Functions ◽  
Class Of Functions ◽  
Symmetrical Points

LetP[A,B],−1≤B<A≤1, be the class of functionspanalytic in the unit diskEwithp(0)=1and subordinate to1+Az1+Bz. In this paper we define and study the classesSS*[A,B]of functions starlike with respect to symmetrical points. A functionfanalytic inEand given byf(z)=z+∑n=2∞anznis said to be inSS*[A,B]if and only if, forz∈E,2zf′(z)f(z)−f(−z)∈P[A,B]. Basic results onSS*[A,B]are studied such as coefficient bounds, distortion and rotation theorems, the analogue of the Polya-Schoenberg conjecture and others.

scholarly journals On certain classes of close-to-convex functions

1993 ◽  
Vol 16 (2) ◽  
pp. 329-336 ◽  
Author(s):  
Khalida Inayat Noor
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
Hadamard Product ◽  
Univalent Functions ◽  
Integral Operators ◽  
Closure Properties ◽  
The Family

A functionf, analytic in the unit diskEand given by ,f(z)=z+∑k=2∞anzkis said to be in the familyKnif and only ifDnfis close-to-convex, whereDnf=z(1−z)n+1∗f,n∈N0={0,1,2,…}and∗denotes the Hadamard product or convolution. The classesKnare investigated and some properties are given. It is shown thatKn+1⫅KnandKnconsists entirely of univalent functions. Some closure properties of integral operators defined onKnare given.

scholarly journals A new general idea for starlike and convex functions

2016 ◽  
Vol 47 (4) ◽  
pp. 445-454 ◽  
Author(s):  
Shigeyoshi Owa ◽  
Srivastava Hari Mohan ◽  
Toshio Hayami ◽  
Kazuo Kuroki
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
General Class ◽  
General Idea ◽  
Open Unit ◽  
Open Unit Disk ◽  
Starlike And Convex Functions ◽  
Class Of Functions

Let $\mathcal{A}$ be the class of functions $f(z)$ which are analytic in the open unit disk $\mathbb{U}$ with $f(0)=0$ and $f'(0)=1$. For the class $\mathcal{A}$, a new general class $\mathcal{A}_{k}$ is defined. With this general class $\mathcal{A}_{k}$, two interesting classes $\mathcal{S}_{k}^{\ast}(\alpha)$ and $\mathcal{K}_{k}(\alpha)$ concerning classes of starlike of order $\alpha$ in $\mathbb{U}$ and convex of order $\alpha$ in $\mathbb{U}$ are considered.

scholarly journals Certain Properties of a Generalized Class of Analytic Functions Involving Some Convolution Operator

2021 ◽  
pp. 49-76
Author(s):  
Faroze Ahmad Malik ◽  
Nusrat Ahmed Dar ◽  
Chitaranjan Sharma
Keyword(s):  
Convex Functions ◽  
Convolution Operator ◽  
Analytic Functions ◽  
Extreme Points ◽  
Integral Operators ◽  
Integral Means ◽  
Coefficient Bounds ◽  
Complex Order ◽  
Special Cases ◽  
The Family

We use the concept of convolution to introduce and study the properties of a unified family $\mathcal{TUM}_\gamma(g,b,k,\alpha)$, $(0\leq\gamma\leq1,\,k\geq0)$, consisting of uniformly $k$-starlike and $k$-convex functions of complex order $b\in\mathbb{C}\setminus\{0\}$ and type $\alpha\in[0,1)$. The family $\mathcal{TUM}_\gamma(g,b,k,\alpha)$ is a generalization of several other families of analytic functions available in literature. Apart from discussing the coefficient bounds, sharp radii estimates, extreme points and the subordination theorem for this family, we settle down the Silverman's conjecture for integral means inequality. Moreover, invariance of this family under certain well-known integral operators is also established in this paper. Some previously known results are obtained as special cases.

scholarly journals On the Fekete-Szegö Problem for a Class of Analytic Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Zhigang Peng
Keyword(s):  
Power Series ◽  
Real Number ◽  
Unit Disk ◽  
Convex Functions ◽  
Analytic Functions ◽  
Upper Bounds ◽  
The Family ◽  
Class Of Functions

Let 𝒜 denote the class of functions which are analytic in the unit disk D={z:|z|<1} and given by the power series f(z)=z+∑n=2∞‍anzn. Let C be the class of convex functions. In this paper, we give the upper bounds of |a3-μa22| for all real number μ and for any f(z) in the family 𝒱={f(z):f∈𝒜, Re(f(z)/g(z))>0 for  some g∈C}.

Subclasses of Starlike Functions Subordinate to Convex Functions

1985 ◽  
Vol 37 (1) ◽  
pp. 48-61 ◽  
Author(s):  
H. Silverman ◽  
E. M. Silvia
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
Starlike Functions ◽  
Schwarz Function ◽  
Image Position ◽  
Class Of Functions

Let S denote the class of functions of the formthat are analytic and univalent in the unit disk Δ = {z:|z| < 1}, with S*(α) and K(α) designating the subclasses of S that are, respectively, starlike of order a and convex of order α, 0 ≦ α < 1. If f(z) and g(z) are analytic in Δ, we say that f(z) is subordinate to g(z), written f ≺ g, if there exists a Schwarz function w(z), w(0) = 0 and |w(z)| < 1 in Δ, such that f(z) = g(w(z)). A function f(z) = z + … is said to be in S*[A, B] if(1)and in K[A, B] if(2)

scholarly journals A Generalized Class of Close-to-Convex Functions

2020 ◽  
Vol 44 (4) ◽  
pp. 533-538
Author(s):  
PARDEEP KAUR ◽  
SUKHWINDER SINGH BILLING
Keyword(s):  
Real Number ◽  
Unit Disk ◽  
Convex Functions ◽  
Starlike Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
Real Numbers ◽  
Special Cases ◽  
Class Of Functions

Let ℋαϕ(β) denote the class of functions f, analytic in the open unit disk ???? which satisfy the condition ( ( ) ) zf-′(z-)- zf-′′(z-) ℜ (1 − α) + α 1 + ′ > β, z ∈ ????, ϕ(z ) f (z ) where α, β are pre-assigned real numbers and ϕ(z) is a starlike function. The special cases of the class ℋαϕ(β) have been studied in literature by different authors. In 2007, Singh et al. [?] studied the class ℋαz(β) and they established that functions in ℋαz(β) are univalent for all real numbers α, β satisfying the condition α ≤ β < 1 and the result is sharp in the sense that constant β cannot be replaced by a real number smaller than α. Singh et al. [?] in 2005, proved that for 0 < α < 1 functions in class ℋαz(α) are univalent. In 1975, Al-Amiri and Reade [?] showed that functions in class ℋαz(0) are univalent for all α ≤ 0 and also for α = 1 in ????. In the present paper, we prove that members of the class ℋαϕ(β) are close-to-convex and hence univalent for real numbers α, β and for a starlike function ϕ satisfying the condition β + α − 1 < αℜ( ) zϕ′(z) ϕ(z)≤ β < 1.

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