scholarly journals The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions of Order 1 / 2

2018 ◽  
Vol 13 (5) ◽  
pp. 2231-2238 ◽  
Author(s):  
Adam Lecko ◽  
Young Jae Sim ◽  
Barabara Śmiarowska
Keyword(s):  
Starlike Functions ◽  
Sharp Bound ◽  
Hankel Determinant ◽  
The Third

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 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.

scholarly journals The Bound of the Hankel Determinant of the Third Kind for Starlike Functions

2018 ◽  
Vol 42 (2) ◽  
pp. 767-780 ◽  
Author(s):  
Oh Sang Kwon ◽  
Adam Lecko ◽  
Young Jae Sim
Keyword(s):  
Starlike Functions ◽  
Hankel Determinant ◽  
The Third

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 Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 347 ◽  
Author(s):  
Shahid Mahmood ◽  
Hari Srivastava ◽  
Nazar Khan ◽  
Qazi Ahmad ◽  
Bilal Khan ◽  
...  
Keyword(s):  
Upper Bound ◽  
Starlike Functions ◽  
Hankel Determinant ◽  
Derivative Operator ◽  
The Third ◽  
Special Cases

The main purpose of this article is to find the upper bound of the third Hankel determinant for a family of q-starlike functions which are associated with the Ruscheweyh-type q-derivative operator. The work is motivated by several special cases and consequences of our main results, which are pointed out herein.

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 ) .

Upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with the q-exponential function

2021 ◽  
Vol 167 ◽  
pp. 102942
Author(s):  
H.M. Srivastava ◽  
Bilal Khan ◽  
Nazar Khan ◽  
Muhammad Tahir ◽  
Sarfraz Ahmad ◽  
...  
Keyword(s):  
Exponential Function ◽  
Upper Bound ◽  
Starlike Functions ◽  
Hankel Determinant ◽  
The Third

scholarly journals The sharp bound of the third Hankel determinant for functions of bounded turning

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Bogumiła Kowalczyk ◽  
Adam Lecko
Keyword(s):  
Analytic Functions ◽  
Sharp Bound ◽  
Hankel Determinant ◽  
The Third ◽  
Bounded Turning

AbstractWe find the sharp bound for the third Hankel determinant $$\begin{aligned} H_{3,1}(f):= \left| {\begin{array}{*{20}c} {a_{1} } & {a_{2} } & {a_{3} } \\ {a_{2} } & {a_{3} } & {a_{4} } \\ {a_{3} } & {a_{4} } & {a_{5} } \\ \end{array} } \right| \end{aligned}$$ H 3 , 1 ( f ) : = a 1 a 2 a 3 a 2 a 3 a 4 a 3 a 4 a 5 for analytic functions f with $$a_n:=f^{(n)}(0)/n!,\ n\in \mathbb N,\ a_1:=1,$$ a n : = f ( n ) ( 0 ) / n ! , n ∈ N , a 1 : = 1 , such that $$\begin{aligned} {{\,\mathrm{Re}\,}}f'(z)>0,\quad z\in \mathbb D:=\{z \in \mathbb C: |z|<1\}. \end{aligned}$$ Re f ′ ( z ) > 0 , z ∈ D : = { z ∈ C : | z | < 1 } .

scholarly journals Coefficient bounds for certain subclasses of starlike functions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Nak Eun Cho ◽  
Virendra Kumar ◽  
Oh Sang Kwon ◽  
Young Jae Sim
Keyword(s):  
Analytic Function ◽  
Upper Bound ◽  
Starlike Functions ◽  
Sharp Bound ◽  
Hankel Determinant ◽  
Coefficient Bounds ◽  
Second Hankel Determinant

Abstract The conjecture proposed by Raina and Sokòł [Hacet. J. Math. Stat. 44(6):1427–1433 (2015)] for a sharp upper bound on the fourth coefficient has been settled in this manuscript. An example is constructed to show that their conjectures for the bound on the fifth coefficient and the bound related to the second Hankel determinant are false. However, the correct bound for the latter is stated and proved. Further, a sharp bound on the initial coefficients for normalized analytic function f such that $zf'(z)/f(z)\prec \sqrt{1+\lambda z}$ z f ′ ( z ) / f ( z ) ≺ 1 + λ z , $\lambda \in (0, 1]$ λ ∈ ( 0 , 1 ] , have also been obtained, which contain many existing results.

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.

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