On the Third and Fourth Hankel Determinants for a Subclass of Analytic Functions

2021 ◽  
Author(s):  
Zhi-Gang Wang ◽  
Mohsan Raza ◽  
Muhammad Arif ◽  
Khurshid Ahmad
Keyword(s):  
Analytic Functions ◽  
Hankel Determinants ◽  
The Third

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.

scholarly journals The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions with Real Coefficients

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 721 ◽  
Author(s):  
Oh Sang Kwon ◽  
Young Jae Sim
Keyword(s):  
Analytic Functions ◽  
Starlike Functions ◽  
Sharp Bound ◽  
Hankel Determinant ◽  
The Third

Let SR * be the class of starlike functions with real coefficients, i.e., the class of analytic functions f which satisfy the condition f ( 0 ) = 0 = f ′ ( 0 ) − 1 , Re { z f ′ ( z ) / f ( z ) } > 0 , for z ∈ D : = { z ∈ C : | z | < 1 } and a n : = f ( n ) ( 0 ) / n ! is real for all n ∈ N . In the present paper, it is obtained that the sharp inequalities − 4 / 9 ≤ H 3 , 1 ( f ) ≤ 3 / 9 hold for f ∈ SR * , where H 3 , 1 ( f ) is the third Hankel determinant of order 3 defined by H 3 , 1 ( f ) = a 3 ( a 2 a 4 − a 3 2 ) − a 4 ( a 4 − a 2 a 3 ) + a 5 ( a 3 − a 2 2 ) .

scholarly journals Bounds on the Third Order Hankel Determinant for Certain Subclasses of Analytic Functions

2017 ◽  
Vol 25 (3) ◽  
pp. 199-214
Author(s):  
S.P. Vijayalakshmi ◽  
T.V. Sudharsan ◽  
Daniel Breaz ◽  
K.G. Subramanian
Keyword(s):  
Series Expansion ◽  
Taylor Series ◽  
Analytic Functions ◽  
Unit Disc ◽  
Taylor Series Expansion ◽  
Upper Bounds ◽  
Hankel Determinant ◽  
Third Order ◽  
The Third

Abstract Let A be the class of analytic functions f(z) in the unit disc ∆ = {z ∈ C : |z| < 1g with the Taylor series expansion about the origin given by f(z) = z+ ∑n=2∞ anzn, z ∈∆ : The focus of this paper is on deriving upper bounds for the third order Hankel determinant H3(1) for two new subclasses of A.

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 The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions with Real Coefficients

2019 ◽  
Author(s):  
Oh Sang Kwon ◽  
Young Jae Sim
Keyword(s):  
Analytic Functions ◽  
Starlike Functions ◽  
Sharp Bound ◽  
Hankel Determinant ◽  
Sharp Estimates ◽  
The Third

Let ${\mathcal{SR}}^*$ be the class of starlike functions with real coefficients, i.e., the class of analytic functions $f$ which satisfy the condition $f(0)=0=f'(0)-1$, Re{z f'(z) / f (z)} &gt; 0, for $z\in\mathbb{D}:=\{z\in\mathbb{C}:|z|&lt;1 \}$ and $a_n:=f^{(n)}(0)/n!$ is real for all $n\in\mathbb{N}$. In the present paper, the sharp estimates of the third Hankel determinant $H_{3,1}$ over the class ${\mathcal{SR}}^*$ are computed.

Questions of generality as probes into nineteenth-century mathematical analysis

2017 ◽  
Author(s):  
Renaud Chorlay
Keyword(s):  
Nineteenth Century ◽  
Set Theory ◽  
Mathematical Analysis ◽  
Analytic Functions ◽  
Function Theory ◽  
The Third ◽  
Point Set

This article examines ways of expressing generality and epistemic configurations in which generality issues became intertwined with epistemological topics, such as rigor, or mathematical topics, such as point-set theory. In this regard, three very specific configurations are discussed: the first evolving from Niels Henrik Abel to Karl Weierstrass, the second in Joseph-Louis Lagrange’s treatises on analytic functions, and the third in Emile Borel. Using questions of generality, the article first compares two major treatises on function theory, one by Lagrange and one by Augustin Louis Cauchy. It then explores how some mathematicians adopted the sophisticated point-set theoretic tools provided for by the advocates of rigor to show that, in some way, Lagrange and Cauchy had been right all along. It also introduces the concept of embedded generality for capturing an approach to generality issues that is specific to mathematics.

scholarly journals Hankel Determinants for New Subclasses of Analytic Functions Related to a Shell Shaped Region

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1041 ◽  
Author(s):  
Gangadharan Murugusundaramoorthy ◽  
Teodor Bulboacă
Keyword(s):  
Power Series ◽  
Upper Bound ◽  
Analytic Functions ◽  
Series Expansions ◽  
Hankel Determinant ◽  
Hankel Determinants ◽  
Power Series Expansions

Using the operator L c a defined by Carlson and Shaffer, we defined a new subclass of analytic functions ML c a ( λ ; ψ ) defined by a subordination relation to the shell shaped function ψ ( z ) = z + 1 + z 2 . We determined estimate bounds of the four coefficients of the power series expansions, we gave upper bound for the Fekete–SzegőSzegő functional and for the Hankel determinant of order two for f ∈ ML c a ( λ ; ψ ) .

scholarly journals A Study of Third Hankel Determinant Problem for Certain Subfamilies of Analytic Functions Involving Cardioid Domain

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 418 ◽  
Author(s):  
Lei Shi ◽  
Izaz Ali ◽  
Muhammad Arif ◽  
Nak Eun Cho ◽  
Shehzad Hussain ◽  
...  
Keyword(s):  
Present Article ◽  
Unit Disk ◽  
Analytic Functions ◽  
Symmetric Functions ◽  
Hankel Determinant ◽  
The Third

In the present article, we consider certain subfamilies of analytic functions connected with the cardioid domain in the region of the unit disk. The purpose of this article is to investigate the estimates of the third Hankel determinant for these families. Further, the same bounds have been investigated for two-fold and three-fold symmetric functions.

scholarly journals Third-Order Hankel Determinant for Certain Class of Analytic Functions Related with Exponential Function

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 501 ◽  
Author(s):  
Hai-Yan Zhang ◽  
Huo Tang ◽  
Xiao-Meng Niu
Keyword(s):  
Unit Disk ◽  
Exponential Function ◽  
Upper Bound ◽  
Analytic Functions ◽  
Function Class ◽  
Open Unit ◽  
Open Unit Disk ◽  
Hankel Determinant ◽  
Third Order ◽  
The Third

Let S l * denote the class of analytic functions f in the open unit disk D = { z : | z | < 1 } normalized by f ( 0 ) = f ′ ( 0 ) − 1 = 0 , which is subordinate to exponential function, z f ′ ( z ) f ( z ) ≺ e z ( z ∈ D ) . In this paper, we aim to investigate the third-order Hankel determinant H 3 ( 1 ) for this function class S l * associated with exponential function and obtain the upper bound of the determinant H 3 ( 1 ) . Meanwhile, we give two examples to illustrate the results obtained.

scholarly journals On the third logarithmic coefficient in some subclasses of close-to-convex functions

2020 ◽  
Vol 114 (2) ◽  
Author(s):  
Nak Eun Cho ◽  
Bogumiła Kowalczyk ◽  
Oh Sang Kwon ◽  
Adam Lecko ◽  
Young Jae Sim
Keyword(s):  
Unit Disk ◽  
Upper Bound ◽  
Convex Functions ◽  
Analytic Functions ◽  
The Third ◽  
Logarithmic Coefficient

AbstractFor analytic functions f in the unit disk $${\mathbb {D}}$$D normalized by $$f(0)=0$$f(0)=0 and $$f'(0)=1$$f′(0)=1 satisfying in $${\mathbb {D}}$$D respectively the conditions $${{\,\mathrm{Re}\,}}\{ (1-z)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z^2)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z+z^2)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z)^2f'(z) \} > 0,$$Re{(1-z)f′(z)}>0,Re{(1-z2)f′(z)}>0,Re{(1-z+z2)f′(z)}>0,Re{(1-z)2f′(z)}>0, the sharp upper bound of the third logarithmic coefficient in case when $$f''(0)$$f′′(0) is real was computed.

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