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.

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 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 ?.

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 Onα-convex functions of orderβ

1997 ◽  
Vol 20 (4) ◽  
pp. 769-772
Author(s):  
Seiichi Fukui
Keyword(s):  
Convex Functions ◽  
Analytic Functions ◽  
New Class

In 1969 Mocanu [1] introduced and studied a new class of analytic functions consisting ofα-convex functions. Many mathematicians have studied and shown the properties of this class. Now we will define new classes like that Mocanu class and then investigate and give some results. The class ofα-convex functions of orderβpartially includes Mocanu's class.

scholarly journals An Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions Associated with k-Fibonacci Numbers

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1043 ◽  
Author(s):  
Muhammad Shafiq ◽  
Hari M. Srivastava ◽  
Nazar Khan ◽  
Qazi Zahoor Ahmad ◽  
Maslina Darus ◽  
...  
Keyword(s):  
Upper Bound ◽  
Analytic Functions ◽  
Fibonacci Numbers ◽  
Starlike Functions ◽  
Function Class ◽  
Hankel Determinant ◽  
Derivative Operator ◽  
New Class ◽  
The Third ◽  
Class A

In this paper, we use q-derivative operator to define a new class of q-starlike functions associated with k-Fibonacci numbers. This newly defined class is a subclass of class A of normalized analytic functions, where class A is invariant (or symmetric) under rotations. For this function class we obtain an upper bound of the third Hankel determinant.

scholarly journals On a Subclass of Analytic Functions Related to a Hyperbola

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Jagannath Patel ◽  
Ashok Kumar Sahoo
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Upper Bound ◽  
Analytic Functions ◽  
Sufficient Condition ◽  
Hankel Determinant ◽  
New Class ◽  
Second Hankel Determinant

The object of the present investigation is to solve Fekete-Szegö problem and determine the sharp upper bound to the second Hankel determinant for a new classℛ̃(a,c,ρ)of analytic functions in the unit disk. We also obtain a sufficient condition for an analytic function to be in this class.

ON HIGHER ORDER BAZILEVIC FUNCTIONS

2012 ◽  
Vol 27 (04) ◽  
pp. 1250203
Author(s):  
KHALIDA INAYAT NOOR ◽  
KHALIL AHMAD
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Higher Order ◽  
Hankel Determinant ◽  
New Class ◽  
Rate Of Growth ◽  
Class B ◽  
Bazilevic Functions

A new class Bk(α, ρ, γ) of analytic functions defined in the unit disc is introduced. This class contains generalized Bazilevic functions of type α. Arclength problem, coefficient and distortion results are studied and also discussed the rate of growth of Hankel determinant for these functions.

An upper bound to the nonlinear functional for certain subclasses of analytic functions associated with Hankel determinant

2014 ◽  
Vol 07 (02) ◽  
pp. 1350042
Author(s):  
D. Vamshee Krishna ◽  
T. Ramreddy
Keyword(s):  
Upper Bound ◽  
Convex Functions ◽  
Analytic Functions ◽  
Hankel Determinant ◽  
Toeplitz Determinants ◽  
Nonlinear Functional ◽  
Determinant Formula ◽  
Starlike And Convex Functions ◽  
Second Hankel Determinant ◽  
Strongly Starlike

The objective of this paper is to obtain an upper bound to the second Hankel determinant [Formula: see text] for the functions belonging to strongly starlike and convex functions of order α(0 < α ≤ 1). Further, we introduce a subclass of analytic functions and obtain the same coefficient inequality for the functions in this class, using Toeplitz determinants.

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