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.

Hankel determinants of second and third order for the class 𝓢 of univalent functions

2021 ◽  
Vol 71 (3) ◽  
pp. 649-654
Author(s):  
Milutin Obradović ◽  
Nikola Tuneski
Keyword(s):  
Unit Disc ◽  
Univalent Functions ◽  
Upper Bounds ◽  
Third Order ◽  
Hankel Determinants

Abstract In this paper we give the upper bounds of the Hankel determinants of the second and third order for the class 𝓢 of univalent functions in the unit disc.

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 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 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 Coefficient Estimates for a Subclass of Starlike Functions

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1646
Author(s):  
Dorina Răducanu
Keyword(s):  
Univalent Functions ◽  
Starlike Functions ◽  
Upper Bounds ◽  
Coefficient Estimates ◽  
Coefficient Problem ◽  
Taylor Coefficients ◽  
The Difference ◽  
Theoretical Results

In this note, we consider a subclass H3/2(p) of starlike functions f with f″(0)=p for a prescribed p∈[0,2]. Usually, in the study of univalent functions, estimates on the Taylor coefficients, Fekete–Szegö functional or Hankel determinats are given. Another coefficient problem which has attracted considerable attention is to estimate the moduli of successive coefficients |an+1|−|an|. Recently, the related functional |an+1−an| for the initial successive coefficients has been investigated for several classes of univalent functions. We continue this study and for functions f(z)=z+∑n=2∞anzn∈H3/2(p), we investigate upper bounds of initial coefficients and the difference of moduli of successive coefficients |a3−a2| and |a4−a3|. Estimates of the functionals |a2a4−a32| and |a4−a2a3| are also derived. The obtained results expand the scope of the theoretical results related with the functional |an+1−an| for various subclasses of univalent functions.

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.

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.

Third Hankel determinants for two classes of analytic functions with real coefficients

2021 ◽  
Vol 33 (4) ◽  
pp. 973-986
Author(s):  
Young Jae Sim ◽  
Paweł Zaprawa
Keyword(s):  
Analytic Functions ◽  
Univalent Functions ◽  
Upper Bounds ◽  
Hankel Determinant ◽  
Lower And Upper Bounds ◽  
Hankel Determinants ◽  
The Third ◽  
Typically Real ◽  
Class Of Functions ◽  
Typically Real Functions

Abstract In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H 2 , 2 {H_{2,2}} or H 3 , 1 {H_{3,1}} over different subclasses of 𝒮 {\mathcal{S}} . Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯 {\mathcal{T}} of typically real functions. The second object of our interest is 𝒦 ℝ ⁢ ( i ) {\mathcal{K}_{\mathbb{R}}(i)} , the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.

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