scholarly journals Fourth Hankel Determinant for a Subclass of Starlike Functions Based on Modified Sigmoid

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Wali Khan Mashwani ◽  
Bakhtiar Ahmad ◽  
Nazar Khan ◽  
Muhammad Ghaffar Khan ◽  
Sama Arjika ◽  
...  
Keyword(s):  
Fourth Order ◽  
Starlike Functions ◽  
Hankel Determinant ◽  
Third Order ◽  
Hankel Determinants ◽  
Sigmoid Functions

In our present investigation, we obtain the improved third-order Hankel determinant for a class of starlike functions connected with modified sigmoid functions. Further, we investigate the fourth-order Hankel determinant, Zalcman conjecture, and also evaluate the fourth-order Hankel determinants for 2-fold, 3-fold, and 4-fold symmetric starlike functions.

scholarly journals A Study of Fourth-Order Hankel Determinants for Starlike Functions Connected with the Sine Function

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Hai-Yan Zhang ◽  
Huo Tang
Keyword(s):  
Fourth Order ◽  
Starlike Functions ◽  
Function Class ◽  
Upper Bounds ◽  
Sine Function ◽  
Hankel Determinant ◽  
Hankel Determinants

In this paper, upper bounds for the fourth-order Hankel determinant H 4 1 for the function class S s ∗ associated with the sine function are given.

scholarly journals Applications of Modified Sigmoid Functions to a Class of Starlike Functions

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Muhammad Ghaffar Khan ◽  
Bakhtiar Ahmad ◽  
Gangadharan Muraugusundaramoorthy ◽  
Ronnason Chinram ◽  
Wali Khan Mashwani
Keyword(s):  
Computer Science ◽  
Special Class ◽  
Symmetric Functions ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Hankel Determinant ◽  
Third Order ◽  
The Third ◽  
Necessary And Sufficient ◽  
Sigmoid Functions

The main focus of this investigation is the applications of modified sigmoid functions. Due to its various uses in physics, engineering, and computer science, we discuss several geometric properties like necessary and sufficient conditions in the form of convolutions for functions to be in the special class S S G ∗ earlier introduced by Goel and Kumar and obtaining third-order Hankel determinant for this class using modified sigmoid functions. Also, the third-order Hankel determinant for 2- and 3-fold symmetric functions of this class is evaluated.

scholarly journals Certain Coefficient Estimate Problems for Three-Leaf-Type Starlike Functions

2021 ◽  
Vol 5 (4) ◽  
pp. 137
Author(s):  
Lei Shi ◽  
Muhammad Ghaffar Khan ◽  
Bakhtiar Ahmad ◽  
Wali Khan Mashwani ◽  
Praveen Agarwal ◽  
...  
Keyword(s):  
Symmetric Functions ◽  
Starlike Functions ◽  
Upper Bounds ◽  
Coefficient Estimate ◽  
Coefficient Estimates ◽  
Third Order ◽  
Hankel Determinants ◽  
Leaf Type ◽  
Logarithmic Coefficients

In our present investigation, some coefficient functionals for a subclass relating to starlike functions connected with three-leaf mappings were considered. Sharp coefficient estimates for the first four initial coefficients of the functions of this class are addressed. Furthermore, we obtain the Fekete–Szegö inequality, sharp upper bounds for second and third Hankel determinants, bounds for logarithmic coefficients, and third-order Hankel determinants for two-fold and three-fold symmetric functions.

scholarly journals Third-Order Hankel and Toeplitz Determinants for Starlike Functions Connected with the Sine Function

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 404 ◽  
Author(s):  
Hai-Yan Zhang ◽  
Rekha Srivastava ◽  
Huo Tang
Keyword(s):  
Starlike Functions ◽  
Function Class ◽  
Open Unit ◽  
Sine Function ◽  
Hankel Determinant ◽  
Third Order ◽  
Toeplitz Determinants ◽  
The Third ◽  
Toeplitz Determinant ◽  
The Right

Let S s * be the class of normalized functions f defined in the open unit disk D = { z : | z | < 1 } such that the quantity z f ′ ( z ) f ( z ) lies in an eight-shaped region in the right-half plane and satisfying the condition z f ′ ( z ) f ( z ) ≺ 1 + sin z ( z ∈ D ) . In this paper, we aim to investigate the third-order Hankel determinant H 3 ( 1 ) and Toeplitz determinant T 3 ( 2 ) for this function class S s * associated with sine function and obtain the upper bounds of the determinants H 3 ( 1 ) and T 3 ( 2 ) .

scholarly journals A study of sharp coefficient bounds for a new subfamily of starlike functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Khalil Ullah ◽  
H. M. Srivastava ◽  
Ayesha Rafiq ◽  
Muhammad Arif ◽  
Sama Arjika
Keyword(s):  
Unit Disk ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Hyperbolic Tangent ◽  
Third Order ◽  
Hyperbolic Tangent Function ◽  
Hankel Determinants ◽  
Coefficient Bounds ◽  
Tangent Function

AbstractIn this article, by employing the hyperbolic tangent function tanhz, a subfamily $\mathcal{S}_{\tanh }^{\ast }$ S tanh ∗ of starlike functions in the open unit disk $\mathbb{D}\subset \mathbb{C}$ D ⊂ C : $$\begin{aligned} \mathbb{D}= \bigl\{ z:z\in \mathbb{C} \text{ and } \vert z \vert < 1 \bigr\} \end{aligned}$$ D = { z : z ∈ C  and  | z | < 1 } is introduced and investigated. The main contribution of this article includes derivations of sharp inequalities involving the Taylor–Maclaurin coefficients for functions belonging to the class $\mathcal{S}_{\tanh }^{\ast } $ S tanh ∗ of starlike functions in $\mathbb{D}$ D . In particular, the bounds of the first three Taylor–Maclaurin coefficients, the estimates of the Fekete–Szegö type functionals, and the estimates of the second- and third-order Hankel determinants are the main problems that are proposed to be studied here.

scholarly journals Coefficient functionals for a class of bounded turning functions related to modified sigmoid function

2021 ◽  
Vol 7 (2) ◽  
pp. 3133-3149
Author(s):  
Muhammad Ghaffar Khan ◽  
◽  
Nak Eun Cho ◽  
Timilehin Gideon Shaba ◽  
Bakhtiar Ahmad ◽  
...  
Keyword(s):  
Present Article ◽  
Fourth Order ◽  
Symmetric Functions ◽  
Second Order ◽  
Sigmoid Function ◽  
Hankel Determinant ◽  
Hankel Determinants ◽  
The Third ◽  
Bounded Turning

<abstract><p>The main objective of the present article is to define the class of bounded turning functions associated with modified sigmoid function. Also we investigate and determine sharp results for the estimates of four initial coefficients, Fekete-Szegö functional, the second-order Hankel determinant, Zalcman conjucture and Krushkal inequality. Furthermore, we evaluate bounds of the third and fourth-order Hankel determinants for the class and for the 2-fold and 3-fold symmetric functions.</p></abstract>

scholarly journals Second Hankel Determinants for Some Subclasses of Biunivalent Functions Associated with Pseudo-Starlike Functions

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
K. Rajya Laxmi ◽  
R. Bharavi Sharma
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Starlike Functions ◽  
Starlike Function ◽  
Open Unit ◽  
Hankel Determinant ◽  
Open Unit Disc ◽  
Hankel Determinants ◽  
Second Hankel Determinant ◽  
Starlike Univalent Function

We introduce second Hankel determinant of biunivalent analytic functions associated with λ-pseudo-starlike function in the open unit disc Δ subordinate to a starlike univalent function whose range is symmetric with respect to the real axis.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

scholarly journals Perturbative linearization of super-Yang-Mills theories in general gauges

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Hannes Malcha ◽  
Hermann Nicolai
Keyword(s):  
Large Class ◽  
Fourth Order ◽  
Landau Gauge ◽  
Third Order ◽  
Yang Mills ◽  
Free Theory ◽  
Non Linear ◽  
Functional Measure ◽  
Non Local ◽  
Jacobi Determinant

Abstract Supersymmetric Yang-Mills theories can be characterized by a non-local and non-linear transformation of the bosonic fields (Nicolai map) mapping the interacting functional measure to that of a free theory, such that the Jacobi determinant of the transformation equals the product of the fermionic determinants obtained by integrating out the gauginos and ghosts at least on the gauge hypersurface. While this transformation has been known so far only for the Landau gauge and to third order in the Yang-Mills coupling, we here extend the construction to a large class of (possibly non-linear and non-local) gauges, and exhibit the conditions for all statements to remain valid off the gauge hypersurface. Finally, we present explicit results to second order in the axial gauge and to fourth order in the Landau gauge.

The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛*

2020 ◽  
Vol 70 (4) ◽  
pp. 849-862
Author(s):  
Shagun Banga ◽  
S. Sivaprasad Kumar
Keyword(s):  
Triangle Inequality ◽  
Starlike Functions ◽  
Sharp Bound ◽  
The Novel ◽  
Sharp Bounds ◽  
Hankel Determinants ◽  
Lemniscate Of Bernoulli ◽  
Carathéodory Functions ◽  
The Right

AbstractIn this paper, we use the novel idea of incorporating the recently derived formula for the fourth coefficient of Carathéodory functions, in place of the routine triangle inequality to achieve the sharp bounds of the Hankel determinants H3(1) and H2(3) for the well known class 𝓢𝓛* of starlike functions associated with the right lemniscate of Bernoulli. Apart from that the sharp bound of the Zalcman functional: $\begin{array}{} |a_3^2-a_5| \end{array}$ for the class 𝓢𝓛* is also estimated. Further, a couple of interesting results of 𝓢𝓛* are also discussed.

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