scholarly journals Some Properties of a Generalized Class of Analytic Functions Related with Janowski Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
M. Arif ◽  
K. I. Noor ◽  
M. Raza ◽  
W. Haq
Keyword(s):  
Convex Functions ◽  
Analytic Functions ◽  
Arc Length ◽  
Hankel Determinant ◽  
Janowski Functions ◽  
Coefficient Problems ◽  
Number Of Classes

We define a classT̃k[A, B,α,ρ] of analytic functions by using Janowski’s functions which generalizes a number of classes studied earlier such as the class of strongly close-to-convex functions. Some properties of this class, including arc length, coefficient problems, and a distortion result, are investigated. We also discuss the growth of Hankel determinant problem.

scholarly journals On Coefficient Problems for Functions Connected with the Sine Function

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1179
Author(s):  
Katarzyna Tra̧bka-Wiȩcław
Keyword(s):  
Analytic Functions ◽  
Sine Function ◽  
Hankel Determinant ◽  
Coefficient Problems ◽  
Logarithmic Coefficients ◽  
Symmetric Points

In this paper, some coefficient problems for starlike analytic functions with respect to symmetric points are considered. Bounds of several coefficient functionals for functions belonging to this class are provided. The main aim of this paper is to find estimates for the following: coefficients, logarithmic coefficients, some cases of the generalized Zalcman coefficient functional, and some cases of the Hankel determinant.

THE SHARP BOUND FOR THE HANKEL DETERMINANT OF THE THIRD KIND FOR CONVEX FUNCTIONS

2018 ◽  
Vol 97 (3) ◽  
pp. 435-445 ◽  
Author(s):  
BOGUMIŁA KOWALCZYK ◽  
ADAM LECKO ◽  
YOUNG JAE SIM
Keyword(s):  
Convex Functions ◽  
Analytic Functions ◽  
Sharp Inequality ◽  
Sharp Bound ◽  
Hankel Determinant ◽  
The Third ◽  
Image Position

We prove the sharp inequality $|H_{3,1}(f)|\leq 4/135$ for convex functions, that is, for analytic functions $f$ with $a_{n}:=f^{(n)}(0)/n!,~n\in \mathbb{N}$, such that $$\begin{eqnarray}Re\bigg\{1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\bigg\}>0\quad \text{for}~z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\},\end{eqnarray}$$ where $H_{3,1}(f)$ is the third Hankel determinant $$\begin{eqnarray}H_{3,1}(f):=\left|\begin{array}{@{}ccc@{}}a_{1} & a_{2} & a_{3}\\ a_{2} & a_{3} & a_{4}\\ a_{3} & a_{4} & a_{5}\end{array}\right|.\end{eqnarray}$$

scholarly journals Uniformly Alpha-Quasi-Convex Functions Defined by Janowski Functions

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Shahid Mahmood ◽  
Sarfraz Nawaz Malik ◽  
Sumbal Farman ◽  
S. M. Jawwad Riaz ◽  
Shabieh Farwa
Keyword(s):  
Integral Representation ◽  
Convex Functions ◽  
Analytic Functions ◽  
Janowski Functions ◽  
Convolution Properties ◽  
Inclusion Results

In this work, we aim to introduce and study a new subclass of analytic functions related to the oval and petal type domain. This includes various interesting properties such as integral representation, sufficiency criteria, inclusion results, and the convolution properties for newly introduced class.

scholarly journals On quasi-convex functions and related topics

1987 ◽  
Vol 10 (2) ◽  
pp. 241-258 ◽  
Author(s):  
Khalida Inayat Noor
Keyword(s):  
Convex Functions ◽  
Unit Disc ◽  
Arc Length ◽  
Basic Properties ◽  
Complete Study ◽  
Coefficient Problems ◽  
Class Of Functions

LetSbe the class of functionsfwhich are analytic and univalent in the unit discEwithf(0)=0,f′(0)=1. LetC,S*andKbe the classes of convex, starlike and close-to-convex functions respectively. The classC*of quasi-convex functions is defined as follows:Letfbe analytic inEandf(0),f′(0)=1. Thenf ϵ C*if and only if there exists ag ϵ Csuch that, forz ϵ ERe(zf′(z))′g′(z)>0.In this paper, an up-to-date complete study of the classC*is given. Its basic properties, its relationship with other subclasses ofS, coefficient problems, arc length problem and many other results are included in this study. Some related classes are also defined and studied in some detail.

scholarly journals On subclasses of close-to-convex functions of higher order

1992 ◽  
Vol 15 (2) ◽  
pp. 279-289 ◽  
Author(s):  
Khalida Inayat Noor
Keyword(s):  
Convex Functions ◽  
Analytic Functions ◽  
Higher Order ◽  
Arc Length ◽  
Covering Theorem ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation

The classesTk(ρ),0≤ρ<1,k≥2, of analytic functions, using the classVk(ρ)of functions of bounded boundary rotation, are defined and it is shown that the functions in these classes are close-to-convex of higher order. Covering theorem, arc-length result and some radii problems are solved. We also discuss some properties of the classVk(ρ)including distortion and coefficient results.

scholarly journals On the second hankel determinant for the kth-root transform of analytic functions

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 227-245 ◽  
Author(s):  
Najla Alarifi ◽  
Rosihan Ali ◽  
V. Ravichandran
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Convex Functions ◽  
Analytic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Hankel Determinant ◽  
Second Hankel Determinant ◽  
The Given ◽  
Logarithmically Convex Functions

Let f be a normalized analytic function in the open unit disk of the complex plane satisfying zf'(z)/f(z) is subordinate to a given analytic function ?. A sharp bound is obtained for the second Hankel determinant of the kth-root transform z[f(zk)/zk]1/k. Best bounds for the Hankel determinant are also derived for the kth-root transform of several other classes, which include the class of ?-convex functions and ?-logarithmically convex functions. These bounds are expressed in terms of the coefficients of the given function ?, and thus connect with earlier known results for particular choices of ?.

On the Radius of Curvature for Convex Analytic Functions

1970 ◽  
Vol 22 (3) ◽  
pp. 486-491 ◽  
Author(s):  
Paul Eenigenburg
Keyword(s):  
Real Axis ◽  
Jordan Curve ◽  
Convex Functions ◽  
Analytic Functions ◽  
Tangent Line ◽  
Positive Real Axis ◽  
Radius Of Curvature ◽  
Arc Length ◽  
Positive Real ◽  
Image Position

Definition 1.1. Let be analytic for |z| < 1. If ƒ is univalent, we say that ƒ belongs to the class S.Definition 1.2. Let ƒ ∈ S, 0 ≦ α < 1. Then ƒ belongs to the class of convex functions of order α, denoted by Kα, provided(1)and if > 0 is given, there exists Z0, |Z0| < 1, such thatLet ƒ ∈ Kα and consider the Jordan curve ϒτ = ƒ(|z| = r), 0 < r < 1. Let s(r, θ) measure the arc length along ϒτ; and let ϕ(r, θ) measure the angle (in the anti-clockwise sense) that the tangent line to ϒτ at ƒ(reiθ) makes with the positive real axis.

scholarly journals Hankel Determinant Problem for q-strongly Close-to-Convex Functions

2022 ◽  
pp. 227-236
Author(s):  
Khalida Inayat Noor ◽  
Muhammad Aslam Noor
Keyword(s):  
Convex Functions ◽  
Analytic Functions ◽  
Hankel Determinant ◽  
New Class

In this paper, we introduce a new class $K_{q}(\alpha), \quad 0<\alpha \leq1, \quad 0<q<1, $ of normalized analytic functions $f $ such that $\big|\arg\frac{D_qf(z)}{D_qg(z)}\big| \leq \alpha \frac{\pi}{2},$ where $g$ is convex univalent in $E= \{z: |z|<1\} $ and $D_qf $ is the $q$-derivative of $f $ defined as: $$D_qf(z)= \frac{f(z)-f(qz)}{(1-q)z}, \quad z\neq0\quad D_qf(0)= f^{\prime}(0). $$ The problem of growth of the Hankel determinant $H_n(k) $ for the class $K_q(\alpha) $ is investigated. Some known interesting results are pointed out as applications of the main results.

scholarly journals Coefficient problems for classes Hq (φ) and Lq (φ)

2021 ◽  
Vol 36 ◽  
pp. 04003
Author(s):  
Aini Janteng ◽  
Andy Pik Hern Liew
Keyword(s):  
Convex Functions ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Upper Bounds ◽  
Derivative Operator ◽  
Coefficient Problems

A class of analytic functions is denoted by M. Furthermore, S⸦M includes of analytic, normalized and univalent functions. The main -subclasses of S are starlike functions, S and convex functions, S* . Recently, many mathematicians studied about the q-derivative operator. Inspired by the ideas from some previous works, we introduce another two new subclasses of M . The coefficient problems in particular the upper bounds of the Fekete-Szegö (F-S) functional for these subclasses were obtained.

scholarly journals Bounds for the Second Hankel Determinant of Certain Univalent Functions

2019 ◽  
Vol 8 (10S) ◽  
pp. 262-266
Keyword(s):  
Convex Functions ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Hankel Determinant ◽  
Starlike And Convex Functions ◽  
Second Hankel Determinant

We study the estimates for the Second Hankel determinant of analytic functions. Our class includes (j,k)-convex, (j,k)-starlike functions and Ma-Minda starlike and convex functions..

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