Milutin Obradović, Nikola Tuneski, Pawel Zaprawa
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.
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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 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.
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。
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 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。
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。
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.
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。
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。
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 $f(z)=z+a_2z^2+\cdots$.
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。
Improvements of certain results of the class $\mathcal{S}$ of univalent functions
本文针对单位圆盘内单叶函数类 $\mathcal{S}$，在 $a_2=0$ 或 $a_3=0$ 条件下改进了第二、第三 Hankel 行列式的上界，并改进了系数差 $|a_4|-|a_3|$ 的上界。
On the symmetric $q$-analog on the bi-univalent functions with respect to symmetric points
本研究引入并探究了与对称$q$-导数算子、广义Bernardi积分算子相关的双单叶函数新子类，推导了约简子类的第二类Hankel行列式、Fekete-Szegö泛函不等式及系数上界。
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。
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.
Results on Logarithmic Coefficients for the Class of Bounded Turning Functions
本文针对豆形域相关的有界转向函数类，确定了对数系数、对数逆系数、第二Hankel行列式及广义Zalcman猜想不等式的精确界。
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。
Sharp inequalities for Logarithmic Coefficients for Certain Classes of Univalent Functions
研究单叶函数类\(\mathcal{S}\)中函数对数系数的托普利兹行列式尖锐界，涵盖\(\alpha\)阶星形、凸、强星形、强凸及有界转向函数子类，还涉及相关逆函数对数系数的托普利兹行列式，给出主要结果及特殊情况的精确界。
Inverse Logarithmic Coefficients, Differences, Hankel Determinant, and Fekete--Szegö Functionals for the Class $\mathcal{C}_e$
本文研究了类$\mathcal{C}_e$中与反向对数系数相关的性质，推导了初始反向对数系数的精确界，并证明了反向对数系数的界，同时给出了与反向对数系数相关的二阶汉克尔行列式的精确界，以及在该类中泛化费克特-塞戈函数的精确上下界。
Logarithmic coefficients for exponential classes of starlike and convex functions
本文研究与指数映射相关的两个解析单叶函数子类，获得尖锐系数估计、对数系数不等式以及Hankel和Toeplitz行列式的界。
Coefficient bounds for starlike functions associated with Gregory coefficients
该文研究与Gregory系数相关的星形函数的系数问题，得到其对数系数的第二Hankel行列式精确上界为1/64，并确定了Fekete-Szegö不等式、Zalcman泛函及广义Zalcman泛函的精确性。
Coefficient Problems and Sharp Determinant Estimates for the Class $\mathcal{P}^{*}$
研究归一化解析函数子类\(\mathcal{S}^*_{q_1}\)的系数问题，通过从属条件定义该子类，得到对数和逆对数系数及其绝对差的精确界，以及相关二阶行列式的精确估计，扩展和细化了已知解析函数子类的界。
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