scholarly journals Hankel, Toeplitz, and Hermitian-Toeplitz Determinants for Certain Close-to-convex Functions

2022 ◽  
Vol 19 (1) ◽  
Author(s):  
Vasudevarao Allu ◽  
Adam Lecko ◽  
Derek K. Thomas
Keyword(s):  
Convex Functions ◽  
Sharp Bound ◽  
Hankel Determinant ◽  
Toeplitz Determinants ◽  
Sharp Bounds ◽  
Second Hankel Determinant

AbstractLet f be analytic in $$\mathbb {D}=\{z\in \mathbb {C}:|z|<1\}$$ D = { z ∈ C : | z | < 1 } , and be given by $$f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$$ f ( z ) = z + ∑ n = 2 ∞ a n z n . We give sharp bounds for the second Hankel determinant, some Toeplitz, and some Hermitian-Toeplitz determinants of functions in the class of Ozaki close-to-convex functions, together with a sharp bound for the Zalcman functional $$J_{2,3}(f).$$ J 2 , 3 ( f ) .

scholarly journals The second Hankel determinant for strongly convex and Ozaki close-to-convex functions

2021 ◽  
Author(s):  
Young Jae Sim ◽  
Adam Lecko ◽  
Derek K. Thomas
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
Univalent Functions ◽  
Inverse Function ◽  
Invariance Property ◽  
Hankel Determinant ◽  
Sharp Bounds ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Second Hankel Determinant

AbstractLet f be analytic in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$ D = { z ∈ C : | z | < 1 } , and $${{\mathcal {S}}}$$ S be the subclass of normalized univalent functions given by $$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$ f ( z ) = z + ∑ n = 2 ∞ a n z n for $$z\in {\mathbb {D}}$$ z ∈ D . We give sharp bounds for the modulus of the second Hankel determinant $$ H_2(2)(f)=a_2a_4-a_3^2$$ H 2 ( 2 ) ( f ) = a 2 a 4 - a 3 2 for the subclass $$ {\mathcal F_{O}}(\lambda ,\beta )$$ F O ( λ , β ) of strongly Ozaki close-to-convex functions, where $$1/2\le \lambda \le 1$$ 1 / 2 ≤ λ ≤ 1 , and $$0<\beta \le 1$$ 0 < β ≤ 1 . Sharp bounds are also given for $$|H_2(2)(f^{-1})|$$ | H 2 ( 2 ) ( f - 1 ) | , where $$f^{-1}$$ f - 1 is the inverse function of f. The results settle an invariance property of $$|H_2(2)(f)|$$ | H 2 ( 2 ) ( f ) | and $$|H_2(2)(f^{-1})|$$ | H 2 ( 2 ) ( f - 1 ) | for strongly convex functions.

INVARIANCE OF THE COEFFICIENTS OF STRONGLY CONVEX FUNCTIONS

2016 ◽  
Vol 95 (3) ◽  
pp. 436-445 ◽  
Author(s):  
D. K. THOMAS ◽  
SARIKA VERMA
Keyword(s):  
Convex Functions ◽  
Classical Result ◽  
Inverse Function ◽  
Hankel Determinant ◽  
Sharp Bounds ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Second Hankel Determinant

Let the function $f$ be analytic in $\mathbb{D}=\{z:|z|<1\}$ and given by $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$. For $0<\unicode[STIX]{x1D6FD}\leq 1$, denote by ${\mathcal{C}}(\unicode[STIX]{x1D6FD})$ the class of strongly convex functions. We give sharp bounds for the initial coefficients of the inverse function of $f\in {\mathcal{C}}(\unicode[STIX]{x1D6FD})$, showing that these estimates are the same as those for functions in ${\mathcal{C}}(\unicode[STIX]{x1D6FD})$, thus extending a classical result for convex functions. We also give invariance results for the second Hankel determinant $H_{2}=|a_{2}a_{4}-a_{3}^{2}|$, the first three coefficients of $\log (f(z)/z)$ and Fekete–Szegö theorems.

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.

scholarly journals Second Hankel Determinant of Logarithmic Coefficients of Convex and Starlike Functions of Order Alpha

2021 ◽  
Author(s):  
Bogumiła Kowalczyk ◽  
Adam Lecko
Keyword(s):  
Convex Functions ◽  
Starlike Functions ◽  
Hankel Determinant ◽  
Sharp Bounds ◽  
Starlike And Convex Functions ◽  
Second Hankel Determinant ◽  
Logarithmic Coefficients ◽  
Convex And Starlike Functions

AbstractIn the present paper, we found sharp bounds of the second Hankel determinant of logarithmic coefficients of starlike and convex functions of order $$\alpha $$ α .

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 Second Hankel Determinant Problem for a Class of Bi-Close-to-Convex Functions

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 986 ◽  
Author(s):  
Nak Eun Cho ◽  
Ebrahim Analouei Adegani ◽  
Serap Bulut ◽  
Ahmad Motamednezhad
Keyword(s):  
Convex Functions ◽  
Hankel Determinant ◽  
Second Hankel Determinant ◽  
Class C ◽  
Proof Methods

The purpose of the present work is to determine a bound for the functional H 2 ( 2 ) = a 2 a 4 − a 3 2 for functions belonging to the class C Σ of bi-close-to-convex functions. The main result presented here provides much improved estimation compared with the previous result by means of different proof methods than those used by others.

scholarly journals Second Hankel determinant for close-to-convex functions

2017 ◽  
Vol 355 (10) ◽  
pp. 1063-1071 ◽  
Author(s):  
Dorina Răducanu ◽  
Paweł Zaprawa
Keyword(s):  
Convex Functions ◽  
Hankel Determinant ◽  
Second Hankel Determinant

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 Sharp Bounds of the Hermitian Toeplitz Determinants for Some Classes of Close-to-Convex Functions

2021 ◽  
Author(s):  
Adam Lecko ◽  
Barbara Śmiarowska
Keyword(s):  
Lower Bounds ◽  
Convex Functions ◽  
Upper And Lower Bounds ◽  
Toeplitz Determinants ◽  
Sharp Bounds

AbstractSharp upper and lower bounds of the Hermitian Toeplitz determinants of the second and third orders are found for various subclasses of close-to-convex functions.

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