H. Orhan, N. Magesh, J. Yamini
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.
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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.
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.
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 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 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 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.
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 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.
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。
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.
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.
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。
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.
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。
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 ${\mathcal U}(λ)$.
Sharp bounds for second Hankel determinant of logarithmic coefficients for certain classes of univalent functions
本文针对标准化单叶函数类，研究了关于对称点的星形和凸函数的对数系数的第二Hankel行列式，建立了其精确上界不等式，并提供了证明严格不等式成立的示例。
Introducing a new method we give sharp estimates of the Hermitian Toepliz determinants of third order for the class $\mathcal{S}$ of functions univalent in the unit disc. The new approach is also illustrated on some subclasses of the class $\mathcal{S}$.
Toeplitz and symmetric Toeplitz determinants for inverse functions of certain classes of univalent functions
本文研究了某些单值函数类的反函数的 Toeplitz 和对称 Toeplitz 确定性，并改进了先前的结果。
Sharp upper and lower bounds for the second and third order Hermitian-Toepilitz determinants are obtained for some generalized subclasses of starlike and convex functions. Applications of these results are also discussed for several widely known classes.
Second Hankel determinant of logarithmic coefficients of inverse functions in certain classes of univalent functions
本文针对标准化单叶函数类逆函数的对数系数，分别就星形函数、凸函数及阶为1/2的有界转角函数，建立了第二 Hankel 行列式的精确上界不等式。
Let $\mathcal{S}$ denote the class of analytic and univalent functions in $\mathbb{D}:=\{z\in\mathbb{C}:\, |z|<1\}$ of the form $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$. In this paper, we determine sharp estimates for the Toeplitz determinants whose elements are the Taylor coefficients of functions in $\mathcal{S}$ and its certain subclasses. We also discuss similar problems for typically real functions.
On some properties of bi-univalent functions in the unit disc
本文利用Grunsky系数方法，研究了双单值函数初始系数模的上界、相邻初始系数模差、初始对数系数模及第二Hankel行列式。
Symmetric Toeplitz determinants of some classes of univalent functions
本文研究了单叶函数类${\mathcal U}$和${\mathcal S}$在特定$q$和$n$值下的对称Toeplitz行列式$T_{q,n}(f)$的估计问题。
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.
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.
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.
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 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。
On certain applications of grunsky coefficients in the theory of univalent functions
本文综述基于Grunsky系数的方法在无解析刻画的一般单叶函数类中的应用，给出了对数系数、Hankel行列式、反函数系数及系数差等的相关估计，部分为精确估计。
Let ${\mathcal U}^+$ be the class of analytic functions $f$ such that $\frac{z}{f(z)}$ has real and positive coefficients and $f^{-1}$ be its inverse. In this paper we give sharp estimates of the initial coefficients and initial logarithmic coefficients for $f$, as well as, sharp estimates of the second and the third Hankel determinant for $f$ and $f^{-1}$. We also show that the Zalcman conjecture holds for functions $f$ from ${\mathcal U}^+$.
Hankel and Toeplitz determinants of logarithmic coefficients of Inverse functions for certain classes of univalent functions
本文针对关于对称点的星形和凸函数类，建立了其逆函数对数系数的 Hankel 和 Toeplitz 行列式的精确不等式，并通过示例验证了结论。
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。
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.
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.
Some application of Grunsky coefficients in the theory of univalent functions
本文基于Grunsky系数方法，研究了单位圆盘内单叶函数类中广义Zalcman猜想特殊情形、第三对数系数及第二Hankel行列式的上界问题。
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.
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.
For $f\in \mathcal{S}$, the class of normalized functions, analytic and univalent in the unit disk $\mathbb{D}$ and given by $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$ for $z\in \mathbb{D}$, we give an upper bound for the coefficient difference $|a_4|-|a_3|$ when $f\in \mathcal{S}$. This provides an improved bound in the case $n=3$ of Grispan's 1976 general bound $||a_{n+1}|-|a_n||\le 3.61\dots .$ Other coefficients bounds, and bounds for the second and third Hankel determinants when $f\in \mathcal{S}$ are found when either $a_2=0,$ or $a_3=0$.
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。
In this paper, we find the coefficient bounds using symmetric Toeplitz determinants for the functions belonging to the subclass $R(q)$.
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。
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.
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