Hankel Determinant Research Articles

Certain coefficient problems of $\mathcal{S}_{e}^{*}$ and $\mathcal{C}_{e}$

In this current study, we consider the classes $\mathcal{S}^{*}_{e}$ and $\mathcal{C}_e$ to obtain sharp bounds for the third Hankel determinant for functions within these classes. Additionally, we provide estimates for the sixth and seventh coefficients while establishing the fourth-order Hankel determinant as well.

Bounds for the Second Hankel Determinant of a General Subclass of Bi-Univalent Functions

The Hankel determinant, which plays a significant role in the theory of univalent functions, is investigated here in the context of bi-univalent analytic functions. Specifically, this paper is dedicated to deriving an upper-bound estimate for the second-order Hankel determinant for a general subclass of bi-univalent analytic functions that incorporate Gegenbauer polynomials within the unit disk. Through the careful specialization of parameters in our main result, we unveil several novel findings.

Sharp coefficient bounds for a class of symmetric starlike functions involving the balloon shape domain

Recent research has extensively explored classes of starlike and convex functions across various domains. This study introduces a novel class of symmetric starlike functions with respect to symmetric points and associated with the balloon shape domain. We establish the explicit representation of all functions in this class. We determine the sharp bounds for the initial four coefficients, the sharp Fekete-Szegö inequality, and the sharp bound for the second Hankel determinant for every function in the newly defined class. Furthermore, we present the new findings on the inverse and logarithmic coefficients sharp bounds for all functions belonging to this class.

Upper Bounds of the Third Hankel Determinant for Bi-Univalent Functions in Crescent-Shaped Domains

This paper investigates the third Hankel determinant, denoted H3(1), for functions within the subclass RS∑*(λ) of bi-univalent functions associated with crescent-shaped regions φ⦅z=z+1+z2. The primary aim of this study is to establish upper bounds for H3(1). By analyzing functions within this specific geometric context, we derive precise constraints on the determinant, thereby enhancing our understanding of its behavior. Our results and examples provide valuable insights into the properties of bi-univalent functions in crescent-shaped domains and contribute to the broader theory of analytic functions.

Differential Subordination and Coefficient Functionals of Univalent Functions Related to $\cos z$

Differential subordination in the complex plane is the generalization of a differential inequality on the real line. In this paper, we consider two subclasses of univalent functions associated with the trigonometric function $\cos z$. Using some properties of the hypergeometric functions, we determine the sharp estimate on the parameter $\beta$ such that the analytic function $p(z)$ satisfying $p(0)=1$, is subordinate to $\cos z$ when the differential expression $p(z)+\beta z (dp(z)/dz)$ is subordinate to the Janowski function. We compute sharp bounds on coefficient functional Hermitian--Toeplitz determinants of the third and the fourth order with an invariance property for such functions. In addition, we estimate bound on Hankel determinants of the second and the third order.

Hankel and symmetric Toeplitz determinants for Sakaguchi starlike functions

Abstract. In this paper, we consider the class of starlike functions with respect to symmetric points which are also known as Sakaguchi starlike functions. We de- termine best possible bounds on Zalcman conjecture |a_n^2 – a_(2n-1) | and generalized Zalcman conjecture |aman − am+n−1| for n = 2 and n = 4, m = 2, respectively for such functions. Further, we compute estimate on third order and fourth order Hankel determinants. As well, we also obtain estimates on third and fourth symmetric Toeplitz determinants. Mathematics Subject Classification (2010): 30C45, 30C80. Keywords: Starlike function, Sakaguchi starlike functions, Zalcman conjecture, third and forth order Hankel determinants, second, third and fourth order symmetric Toeplitz determinants.

Sharp Results for a New Class of Analytic Functions Associated with the q-Differential Operator and the Symmetric Balloon-Shaped Domain

In our current study, we apply differential subordination and quantum calculus to introduce and investigate a new class of analytic functions associated with the q-differential operator and the symmetric balloon-shaped domain. We obtain sharp results concerning the Maclaurin coefficients the second and third-order Hankel determinants, the Zalcman conjecture, and its generalized conjecture for this newly defined class of q-starlike functions with respect to symmetric points.

Fekete-Szegö and Hankel inequalities related to the sine function

ABSTRACT The Fekete-Szegö inequality is one of the inequalities for the coefficients which associated with the famous Bieberbach conjecture. Other issues associated with this inequality are determining the Hankel determinant denoted as Hd inequalities which are involved in the study of the singularities and integral coefficients in the Taylor series representations. In this investigation, we discuss such inequalities for certain mappings g for which ( g ′ ( z ) ) α [ 2 z g ′ ( z ) g ( z ) − g ( − z ) ] ( 1 − α ) ≺ 1 + sin ⁡ z , ( 0 ≤ α ≤ 1 ) lies in the image domain starlike about 1 and symmetric about real axis. Particularly, we obtain certain inequality for functions involving Poisson distribution. Furthermore, we also find the second Hankel determinant and other inequalities related to this newly defined class.

Third Hankel Determinant for the Class of Analytic Functions Defined by the Mathieu-Type Series Related to a Petal-Shaped Domain

Third Hankel Determinant for the Class of Analytic Functions Defined by the Mathieu-Type Series Related to a Petal-Shaped Domain

A family of integrable maps associated with the Volterra lattice

Recently Gubbiotti, Joshi, Tran and Viallet classified birational maps in four dimensions admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. The last three of the maps so obtained were shown to be Liouville integrable, in the sense of admitting a non-degenerate Poisson bracket with two first integrals in involution. Here we show how the first of these three Liouville integrable maps corresponds to genus 2 solutions of the infinite Volterra lattice, being the g = 2 case of a family of maps associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus g⩾1 . The continued fraction method provides explicit Hankel determinant formulae for tau functions of the solutions, together with an algebro-geometric description via a Lax representation for each member of the family, associating it with an algebraic completely integrable system. In particular, in the elliptic case (g = 1), as a byproduct we obtain Hankel determinant expressions for the solutions of the Somos-5 recurrence, but different to those previously derived by Chang, Hu and Xin. By applying contraction to the Stieltjes fraction, we recover integrable maps associated with Jacobi continued fractions on hyperelliptic curves, that one of us considered previously, as well as the Miura-type transformation between the Volterra and Toda lattices.

Sharp Coefficient Related Results for Nephroid Shaped Domain

In this study, the focus is on two main objectives related to starlike functions associated with a nephroid-shaped domain. Firstly, the aim is to determine sharp bounds for the coefficients of these functions up to the fifth order. These bounds are crucial as they provide a detailed understanding of the behavior of the coefficients, which is important for further analysis and various applications of these functions. The sharp determination of these coefficients can aid in refining mathematical models and theoretical frameworks involving starlike functions. Secondly, the sharp bound for the third order Hankel determinant for functions in this class is also derived. The Hankel determinant is a significant tool in complex analysis, as it provides insights into the growth, distortion, and other important properties of functions. By deriving these sharp bounds, this study improves upon the existing results in the literature, thereby contributing to a more sharp characterization of starlike functions associated with nephroid-shaped domains. This advancement has the potential to lead to enhanced applications, such as in geometric function theory and fluid dynamics, and offers a deeper understanding of these mathematical functions. By addressing these objectives, the study not only fills gaps in the current research but also opens new avenues for future exploration in the field of complex analysis.

The sharp bound of the third Hankel determinant for certain subfamilies of analytic functions

The sharp bound of the third Hankel determinant for certain subfamilies of analytic functions

Hankel and Toeplitz Determinants of Logarithmic Coefficients of Inverse Functions for the Subclass of Starlike Functions with Respect to Symmetric Conjugate Points

This paper focuses on finding the upper bounds of the second Hankel and Toeplitz determinants, whose entries are logarithmic coefficients of inverse functions for a new subclass of starlike functions with respect to symmetric conjugate points associated with the exponential function defined by subordination. Results on initial Taylor coefficients and logarithmic coefficients of inverse functions for a new subclass are also presented. This study may inspire others to focus further to the coefficient functional problems associated with the inverse functions of various classes of univalent functions.

Sharp Coefficient Estimates for Analytic Functions Associated with Lemniscate of Bernoulli

The main purpose of this work is to study the third Hankel determinant for classes of Bernoulli lemniscate-related functions by introducing new subclasses of star-like functions represented by SLλ* and RLλ. In many geometric and physical applications of complex analysis, estimating sharp bounds for problems involving the coefficients of univalent functions is very important because these coefficients describe the fundamental properties of conformal maps. In the present study, we defined sharp bounds for function-coefficient problems belonging to the family of SLλ* and RLλ. Most of the computed bounds are sharp. This study will encourage further research on the sharp bounds of analytical functions related to new image domains.

Upper bound for the second and third Hankel determinants of analytic functions associated with the error function and q-convolution combination

Recently, El-Deeb and Cotîrlă (Mathematics 11:11234834, 2023) used the error function together with a q-convolution to introduce a new operator. By means of this operator the following class Rα,ϒλ,q(δ,η)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{R}_{\\alpha ,\\Upsilon}^{\\lambda ,q}(\\delta ,\\eta )$\\end{document} of analytic functions was studied: Rα,ϒλ,q(δ,η):={F:ℜ((1−δ+2η)Hϒλ,qF(ζ)ζ+(δ−2η)(Hϒλ,qF(ζ))′+ηζ(Hϒλ,qF(ζ))″)}>α(0≦α<1).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} &\\mathcal{R}_{\\alpha ,\\Upsilon }^{\\lambda ,q}(\\delta ,\\eta ) \\\\ &\\quad := \\biggl\\{ \\mathcal{ F}: {\\Re} \\biggl( (1-\\delta +2\\eta ) \\frac{\\mathcal{H}_{\\Upsilon }^{\\lambda ,q}\\mathcal{F}(\\zeta )}{\\zeta}+(\\delta -2\\eta ) \\bigl(\\mathcal{H} _{\\Upsilon}^{\\lambda ,q}\\mathcal{F}(\\zeta ) \\bigr) ^{{ \\prime}}+\\eta \\zeta \\bigl( \\mathcal{H}_{\\Upsilon}^{\\lambda ,q} \\mathcal{F}( \\zeta ) \\bigr) ^{{{\\prime \\prime}}} \\biggr) \\biggr\\} \\\\ &\\quad >\\alpha \\quad (0\\leqq \\alpha < 1). \\end{aligned}$$ \\end{document}For these general analytic functions F∈Rβ,ϒλ,q(δ,η)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{F}\\in \\mathcal{R}_{\\beta ,\\Upsilon}^{\\lambda ,q}(\\delta , \\eta )$\\end{document}, we give upper bounds for the Fekete–Szegö functional and for the second and third Hankel determinants.

Boundary values of Hankel and Toeplitz determinants for [formula omitted]-convex functions

The study of holomorphic functions has been recently extended through the application of diverse techniques, among which quantum calculus stands out due to its wide-ranging applications across various scientific disciplines. In this context, we introduce a novel q-differential operator defined via the generalized binomial series, which leads to the derivation of new classes of quantum-convex (q-convex) functions. Several specific instances within these classes were explored in detail. Consequently, the boundary values of the Hankel determinants associated with these functions were analyzed. All graphical representations and computational analyses were performed using Mathematica 12.0.•These classes are defined by utilizing a new q-differential operator.•The coefficient values |ai|(i=2,3,4) are investigated.•Toeplitz determinants, such as the second T2(2) and the third T3(1) order inequalities, are calculated.

Coefficients Estimates for a Subclass of Starlike Functions

The paper mainly investigates the initial coefficients for the subclasses of starlike functions defined by using the Cosine function involving $\alpha$ ($0\leq\alpha&lt;1$), we obtain upper bounds for initial order of Hankel determinants and symmetric Toeplitz determinants whose elements are the initial coefficients. Also, we obtain initial coefficient estimation of logarithmic coefficients for the subclass.

Sharp Coefficient Bounds for Starlike Functions Associated with Cosine Function

Let Scos* denote the class of normalized analytic functions f in the open unit disk D satisfying the subordination zf′(z)f(z)≺cosz. In the first result of this article, we find the sharp upper bounds for the initial coefficients a3, a4 and a5 and the sharp upper bound for module of the Hankel determinant |H2,3(f)| for the functions from the class Scos*. The next section deals with the sharp upper bounds of the logarithmic coefficients γ3 and γ4. Then, in addition, we found the sharp upper bound for H2,2Ff/2. To obtain these results we utilized the very useful and appropriate Lemma 2.4 of N.E. Cho et al. [Filomat 34(6) (2020), 2061–2072], which gave a most accurate description for the first five coefficients of the functions from the Carathéodory’s functions class, and provided a technique for finding the maximum value of a three-variable function on a closed cuboid. All the maximum found values were checked by using MAPLE™ 2016 computer software, and we also found the extremal functions in each case. All of our most recent results are the best ones and give sharp versions of those recently published in [Hacet. J. Math. Stat. 52, 596–618, 2023].

Joint moments of higher order derivatives of CUE characteristic polynomials II: structures, recursive relations, and applications

In a companion paper (Keating and Wei 2023 Int. Math. Res. Not. 2024 9607–32), we established asymptotic formulae for the joint moments of higher order derivatives of the characteristic polynomials of CUE random matrices. The leading order coefficients of these asymptotic formulae are expressed as partition sums of derivatives of determinants of Hankel matrices involving I-Bessel functions, with column indices shifted by Young diagrams. In this paper, we continue the study of these joint moments and establish more properties for their leading order coefficients, including structure theorems and recursive relations. We also build a connection to a solution of the σ-Painlevé III ′ equation. In the process, we give recursive formulae for the Taylor coefficients of the Hankel determinants formed from I-Bessel functions that appear at zero and find some differential equations these determinants satisfy. The approach we establish is applicable to determinants of general Hankel matrices whose columns are shifted by Young diagrams.

On Some Properties of a Class of Analytic Functions Defined by Opoola Differential Operator

The Fekete-Szego functional upper bounds and the Second Hankel Determinant upper bounds for a class of analytical functions defined by the Opoola Differential Operator are found in this study. The estimations made are accurate with the theorems proved.