Milutin Obradovic, Nikola Tuneski
Let $f$ be analytic in the unit disk $\mathbb D$ and normalized so that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper, we give upper bounds of the Hankel determinant of second order for the classes of starlike functions of order $α$, Ozaki close-to-convex functions and two other classes of analytic functions. Some of the estimates are sharp.
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The objective of this paper is to find the best possible upper bound of the third Hankel determinant for the inverse of convex functions.
In this paper we improve the upper bound of the third order Hankel determinant for the class of Ozaki close-to-convex functions. The sharp bound is conjectured.
In this paper we give improved, probably not sharp, upper bounds of the Hankel determinant of third order for various classes of univalent functions and conjecture the sharp one.
The estimates for the second Hankel determinant a_2a_4-a_3^2 of analytic function f(z)=z+a_2 z^2+a_3 z^3+...b for which either zf'(z)/f(z) or 1+zf"(z)/f'(z) is subordinate to certain analytic function are investigated. The estimates for the Hankel determinant for two other classes are also obtained. In particular, the estimates for the Hankel determinant of strongly starlike, parabolic starlike, lemniscate starlike functions are obtained.
In this paper, we investigate the sharp bounds of the second Hankel determinant of Logarithmic coefficients for the starlike and convex functions with respect to symmetric points in the open unit disk.
In this paper we give the upper bounds of the Hankel determinants of the second and third order for the class $\mathcal{S}$ of univalent functions in the unit disc.
Let f be analytic in D={z:|z|<1} with f(0)=0 and f'(0)=1. We give sharp bounds for the second Hankel determinant of f, when f is starlike of order alpha in D.
In the present work, we propose to investigate the second Hankel determinant inequalities for certain class of analytic and bi-univalent functions. Some interesting applications of the results presented here are also discussed.
The Third Hankel determinant for inverse coefficients of bounded turning functions
本文研究有界旋转函数逆系数的第三Hankel行列式，获得尖锐界，否定一个猜想，并对相关类给出肯定回答。
Second Hankel Determinant for Logarithmic Inverse Coefficients of Convex and Starlike Functions
本文研究了星形和凸函数的对数逆系数的二次Hankel行列式精确界。
In this paper we improve the bounds of the third order Hankel determinant for two classes of univalent functions with bounded turning. The bounds are not sharp, but the sharp ones are conjectured.
Making use of Chebyshev polynomials, we obtain upper bound estimate for the second Hankel determinant of a subclass $\mathcal{N}_{σ}^μ\left( λ,t\right) $ of bi-univalent function class $σ.$
In the present investigation the authors obtain upper bounds for the second Hankel determinant of the classes bi-starlike and bi-convex functions of order beta.
Second Hankel determinant for logarithmic inverse coefficients of strongly convex and strongly starlike functions
本文研究了α阶强星形与强凸函数类中对数逆系数的第二Hankel行列式，并给出了其精确界。
In this paper our purpose is to find upper bound estimate for the second Hankel determinant $|a_{2}a_{4}-a_{3}^{2}|$ for functions defined by convolution belonging to the class $\mathcal{N}_σ^{μ,δ}(λ,t)$ by using Chebyshev polynomials.
In this paper, we consider a general subclass of analytic and bi-univalent functions in the open unit disk in the complex plane. Making use of the Chebyshev polynomials, we obtain upper bound estimate for the second Hankel determinant for this function class.
In this paper we determine the upper bounds of $|H_{2}(3)|$ for the inverse functions of functions of some classes of univalent functions, where $H_{2}(3)(f)=a_{3}a_{5}-a_{4}^{2}$ is the Hankel determinant of a special type.
On Hankel Determinants of $β$-Spirallike Mappings in Complex Banach Spaces
研究复Banach空间单位球上归一化β-螺旋状映射的第二个Hankel行列式，利用Banach空间全纯映射与单变量β-螺旋状函数关系，先建立初始齐次展开系数估计，进而得出第二个Hankel行列式的精确界。
We consider a family of all analytic and univalent functions (i.e., one-to-one) in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper, we obtain the sharp bounds of the second Hankel determinant of Logarithmic coefficients for some subclasses of analytic functions.
In this sequel to the recent work (see Azizi et al., 2015), we investigate a subclass of analytic and bi-univalent functions in the open unit disk. We obtain bounds for initial coefficients, the Fekete-Szegö inequality and the second Hankel determinant inequality for functions belonging to this subclass. We also discuss some new and known special cases, which can be deduced from our results.
In this paper, we consider the class of strongly bi-close-to-convex functions of order $α$ and bi-close-to-convex functions of order $β$. We obtain an upper bound estimate for the second Hankel determinant for functions belonging to these classes. The results in this article improve some earlier result obtained for the class of bi-convex functions.
An improvement of the estimates of the modulus of the Hankel determinants of second and third order for the class $\mathcal{S}$ of univalent functions
本文利用 Grunsky 系数性质，改进了单叶函数类 $\mathcal{S}$ 的二阶与三阶 Hankel 行列式模的上界估计。
Let $f$ be analutic in the unit disk $\mathbb D$ and normalized so that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper we give sharp bound of Hankel determinant of the second order for the class of analytic unctions satisfying \[ \left|\arg \left[\left(\frac{z}{f(z)}\right)^{1+α}f'(z) \right] \right|<γ\fracπ{2} \quad\quad (z\in\mathbb D),\] for $0<α<1$ and $0<γ\leq1$.
In this paper, we obtain the upper bounds to the third Hankel determinants for starlike functions of order $α$, convex functions of order $α$ and bounded turning functions of order $α$. Furthermore, several relevant results on a new subclass of close-to-convex harmonic mappings are obtained. Connections of the results presented here to those that can be found in the literature are also discussed.
In this paper, new class of bi-univalent functions are introduced. Upper bound of the second Hankel determinant $|H_2(2)|$ of subclass of bi-univalant functions class $Σ$, which defined by subordination, investigated. Furthermore, some results concluded as a special case of our main results and corrected some previous researchers results which investigated by miscalculation.
Focus in this paper is on the Hankel determinant, $H_3(1)$, for the well-known classes of bounded-turning, starlike and convex functions in the open unit disk $E=\{z\in \mathbb{C}\colon|z|<1\}$. The results obtained complete the series of research works in the search for sharp upper bounds on $H_3(1)$ for each of these classes.
Hankel determinant for a general subclass of m-fold symmetric bi-univalent functions defined by Ruscheweyh operator
本文利用Hankel行列式和Ruscheweyh导数研究一般m折对称双单叶函数子类，给出第二Hankel行列式的界及推论，并用Python验证准确性。
Sharp Estimates of Hankel Determinants for certain classes of convex univalent functions
本文研究了特定凸单值函数类的Hankel行列式性质，计算了其第二和第三Hankel行列式的精确值，并确定了极值函数。
On estimation of Hankel determinants for certain class of starlike functions
本研究针对与豆形区域相关的两类星形和凸函数子类，分别估计了其精确初始系数及不同阶数的Hankel行列式，为相关函数类的性质研究提供了精确的行列式结果。
The third Hankel determinant for inverse coefficients of starlike functions of order 1/2
本文求出了1/2阶星形函数逆函数系数第三Hankel行列式的精确界，并由此证明两个相关泛函在该函数类上具有不变性。
The aim of the present paper is to obtain the sharp bounds of the Hankel determinants H_2(3) and H_3(1) for the well known class SL^* of starlike functions associated with the right lemniscate of Bernoulli. Further for n=3, we find the sharp bound of the Zalcman functional for the class SL^*. In addition, a couple of interesting results of SL^* is appended at the end.
In this paper we determine the upper bounds of the Hankel determinants of special type $H_{2}(3)(f)$ and $H_{2}(4)(f)$ for the class of univalent functions and for the class $\mathcal{U}$ defined by \[ \mathcal{U}=\left\{ f\in\mathcal{A} : \left|\left[\frac{z}{f(z)}\right]^2 f'(z)-1 \right|<1,\, z\in{\mathbb D} \right\}, \] where $\mathcal{A}$ is the class of functions analytic in the unit disk ${\mathbb D}$ and normalized such that $f(z)=z+a_2z^2+\cdots$.
Sharp Hankel and Hermitian--Toeplitz Determinants Involving Logarithmic and Inverse Logarithmic Coefficients for the class \(\mathcal{S}_{car}^*\)
本文针对与心脏线区域相关的星型函数类，推导了二阶对数系数与逆对数系数的Hankel行列式的sharp界，完整解决了三阶Hermitian–Toeplitz行列式的sharp极值问题，相关结果由极值实例验证。
The second and third Hankel determinants for certain classes of functions
本文研究了特定函数类的第二个和第三个Hankel行列式，并确定了极值函数以确定系数界。
On sharp third Hankel determinant for certain starlike functions
本文推广了Rath等人的结果，给出了阶数$\alpha$在$[0,1/6]\cup\{1/2\}$范围内的星形函数三阶汉克尔行列式的锐利上界估计。
Hankel, Toeplitz, Hermitian--Toeplitz Determinants and the Zalcman Functional for a Class of Biholomorphic Mappings on Complex Banach Spaces
研究复Banach空间上一类双全纯映射的多种行列式及泛函，受单位圆盘解析函数结果启发，建立Banach空间类似物，将单变量不等式推广到高维，为相关类提供对应。
Let ${\mathcal A}$ be the class of functions that are analytic in the unit disc ${\mathbb D}$, normalized such that $f(z)=z+\sum_{n=2}^\infty a_nz^n$, and let class ${\mathcal U}(λ)$, $0<λ\le1$, consists of functions $f\in{\mathcal A}$, such that \[ \left |\left (\frac{z}{f(z)} \right )^{2}f'(z)-1\right | < λ\quad (z\in {\mathbb D}). \] In this paper we determine the sharp upper bounds for the Hankel determinants of second and third order for the inverse functions of functions from the class ${\mathcal U}(λ)$.
Recently, the subclass of starlike functions associated with exponential function $e^z$, given by ${S}^*_e = \{f(z)\in {S}:{zf'(z)}/{f(z)} \prec e^z, (z\in \mathbb{D}) \}$ was introduced and studied by Mendiratta $et$ $al.$ ~\cite{mendiratta2015subclass}. In this article, we obtain sharp bound for third Hankel determinant for the class $S_e^\ast$, by improving the already known results, in this direction. In our methodology, we use the newly obtained expression for the fourth coefficient of Carathéodory functions.
In the present paper, we will discuss the Hankel determinants $H(f) =a_2a_4-a_3^2$ of order 2 for normalized concave functions $f(z)=z+a_2z^2+a_3z^3+\dots$ with a pole at $p\in(0,1).$ Here, a meromorphic function is called concave if it maps the unit disk conformally onto a domain whose complement is convex. To this end, we will characterize the coefficient body of order 2 for the class of analytic functions $φ(z)$ on $|z|<1$ with $|φ|<1$ and $φ(p)=p.$ We believe that this is helpful for other extremal problems concerning $a_2, a_3, a_4$ for normalized concave functions with a pole at $p.$
The sharp bound of the third order Hankel determinant for inverse of Ozaki close-to-convex functions
本研究针对单位圆盘内的Ozaki近凸函数类，推导其反函数的三阶Hankel行列式的精确界，丰富了单叶函数几何理论中相关行列式估计的研究成果。
The second and third Hankel determinants for certain convex subclass of functions
本文研究了特定凸函数子类的第二和第三Hankel行列式，通过分析函数类的性质，得出了精确的行列式结果。
The second and third Hankel determinants for starlike MA--Minda subclass associated to quadratic polynomials
本文研究了与二次多项式相关的星形MA-Minda子类的第二和第三Hankel行列式性质，确定了极限函数以证明界是 sharp 的。
On the Third Hankel Determinant for Inverse Coefficients of Starlike Functions: A Bernstein Polynomial Approach
本文利用伯恩斯坦多项式方法，研究星形单值函数逆系数的第三Hankel行列式上限估计，得出精确上界。
Following the trend of coefficient bound problems in Geometric Function Theory, in the present paper, we obtain the sharp bound of $|H_3(1)|$ for the class $\mathcal{S}^*$, of starlike functions and $\mathcal{SL}_q^*$, of $q$- starlike functions related with lemniscate of Bernoulli. Bound on the initial class is also an improvement over the existing known bound and the bound on the latter class generalizes the prior known outcome. Further, we determine the extremal functions to prove the sharpness of our results.
Coefficient estimates and Hankel determinant for the class of leaf shaped starlike function
本文研究叶形星像函数子类，利用Carathéodory函数方法获得初始Taylor系数、对数系数及多个Hankel行列式的精确界，并确定极值函数。
Second Hankel determinant of Logarithmic coefficients for Starlike and Convex functions associated with lune
本文研究与月牙形相关的星形函数和凸函数的对数系数，建立了其第二类Hankel行列式的两个精确上界不等式。
Using continued fraction expansions of certain polygamma functions as a main tool, we find orthogonal polynomials with respect to the odd-index Bernoulli polynomials $B_{2k+1}(x)$ and the Euler polynomials $E_{2k+ν}(x)$, for $ν=0, 1, 2$. In the process we also determine the corresponding Jacobi continued fractions (or J-fractions) and Hankel determinants. In all these cases the Hankel determinants are polynomials in $x$ which factor completely over the rationals.
We study $n\times n$ Hankel determinants constructed with moments of a Hermite weight with a Fisher-Hartwig singularity on the real line. We consider the case when the singularity is in the bulk and is both of root-type and jump-type. We obtain large $n$ asymptotics for these Hankel determinants, and we observe a critical transition when the size of the jumps varies with $n$. These determinants arise in the thinning of the generalised Gaussian unitary ensembles and in the construction of special function solutions of the Painlevé IV equation.
We obtain asymptotics of large Hankel determinants whose weight depends on a one-cut regular potential and any number of Fisher-Hartwig singularities. This generalises two results: 1) a result of Berestycki, Webb and Wong [5] for root-type singularities, and 2) a result of Its and Krasovsky [37] for a Gaussian weight with a single jump-type singularity. We show that when we apply a piecewise constant thinning on the eigenvalues of a random Hermitian matrix drawn from a one-cut regular ensemble, the gap probability in the thinned spectrum, as well as correlations of the characteristic polynomia…
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