The third logarithmic coefficient for the class S

二次Hankel行列式与凸及星形函数的对数逆系数

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。

强凸函数与强星形函数的对数逆系数的第二Hankel行列式

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 the Logarithmic Coefficients of Close to Convex Functions

For $f$ analytic and close to convex in $D=\{z: |z|< 1\}$, we give sharp estimates for the logarithmic coefficients $γ_{n}$ of $f$ defined by $\log \dfrac{f(z)}{z}=2\sum_{n=1}^{\infty} γ_{n}z^{n}$ when $n=1, 2,3$.

关于单值函数逆函数对数系数的一些性质

On some properties of logarithmic coefficients of inverse of univalent functions

本文研究了单位圆内单值函数逆函数的初始对数系数性质，特别讨论了凸函数情况，给出了某些情况下初始系数模的估计及相邻系数模差的估计。

关于单叶函数对数系数的某些不等式的简单证明

Simple proofs of certain inequalities with logarithmic coefficients of univalent functions

本文针对一般单叶函数类与凸单叶函数类，给出了第二与第一对数系数模之差界限的简单证明，部分界限为精确的。

Second Hankel determinant of logarithmic coefficients of certain analytic functions

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.

A correct proof of logarithmic coefficients for Janowski type $(j,k)$-symmetric starlike functions

Recently, authors [7] studied the logarithmic coefficient bounds for class of the Janowski type $(j,k)$-symmetric starlike functions $\mathcal{ST}_{[j,k]}(A,B)$ in ({\em Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.} (2022). DOI: 10.1007/s13398-022-01310-9). We pointed out that the proof of Theorem~3 in [7] is incorrect. In this article, we present the correct proof of Theorem~3. In addition, we also obtain some new results related to the logarithmic coefficient inequalities for the class $\mathcal{ST}_{[j,k]}(A,B)$.

Toeplitz determinants of Logarithmic coefficients for Starlike and Convex functions

In this study, we deal with the sharp bounds of certain Toeplitz determinants whose entries are the logarithmic coefficients of analytic univalent functions $f$ such that the quantity $z f'(z)/f(z)$ takes values in a specific domain lying in the right half plane. The established results provide the bounds for the classes of starlike and convex functions, as well as various of their subclasses.

关于两个初始对数系数模的差

On the difference of the moduli of the two initial logarithmic coefficients

本文研究几类解析函数中第二与第一对数系数模之差的精确界，覆盖函数类 $\mathcal U$、$α$-凸函数及 Ozaki 类 $\mathcal{G}(α)$。

关于Ma-Minda型凸函数的对数系数

On the logarithmic coefficients of Ma-Minda type convex functions

本文研究了三种特定的Ma-Minda型凸函数子类，推导了关于个体系数估计和加权级数的对数系数不等式，修正了先前错误结果并扩展了已知结论。

对某一类星形函数的对数系数的精确估计

Sharp Estimates of Logarithmic Coefficients for a Certain Class of Starlike Functions

本文研究了与双曲余弦函数相关的星形函数类中对数系数的极值性质，确定了初始对数系数γ₁、γ₂、γ₃的精确上界，并给出了第二Hankel行列式H_{2,1}(F_f/2)的精确界，同时扩展分析到反函数，推导了反函数的对数逆系数及相应行列式的精确估计。

关于单叶函数类初始对数系数之差的研究

On the difference of initial logarithmic coefficients for the class of univalent functions

本文针对单位圆盘内规范化的单叶函数类，给出了初始对数系数模的差值 |γ₃|-|γ₂| 与 |γ₄|-|γ₃| 的估计。

几何闭合凸函数类的对数系数问题

Logarithmic Coefficients Problems of Geometric Subclass of Closed-to-convex Functions

本文研究了几何闭合凸函数类W(α)中对数系数γ₁、γ₂、γ₃以及对数反系数Γ₁、Γ₂、Γ₃的精确界，得出了|γ₂|−|γ₁|和|Γ₂|−|Γ₁|的精确上界和下界，并建立了对数和反对数系数的第二Hankel行列式的精确不等式。

Logarithmic coefficients for certain subclasses of close-to-convex functions

Let $\mathcal{S}$ denote the class of functions analytic and univalent (i.e. one-to-one) in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:\, |z|<1\}$ normalized by $f(0)=0=f'(0)-1$. The logarithmic coefficients $γ_n$ of $f\in\mathcal{S}$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty} γ_n z^n.$ In the present paper, we determine the sharp upper bounds for $|γ_1|$, $|γ_2|$ and $|γ_3|$ when $f$ belongs to some familiar subclasses of close-to-convex functions.

Logarithmic coefficients of some close-to-convex functions

The logarithmic coefficients $γ_n$ of an analytic and univalent function $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty} γ_n z^n$. In the present paper, we consider close-to-convex functions (with argument $0$) with respect to odd starlike functions and determine the sharp upper bound of $|γ_n|$, $n=1,2,3$ for such functions $f$.

某类Sakaguchi函数系数泛函的精确界

Sharp bounds on Coefficient functionals of certain Sakaguchi functions

该研究针对两类与伯努利双纽线右半支及指数函数相关的Sakaguchi函数子类，推导了多类行列式的精确界并探讨其系数不变性。

Janowski类的对数逆系数与模差

Logarithmic Inverse Coefficients and Moduli Differences of Janowski Class

本文研究Janowski凸类C(A,B)中前三个对数逆系数的精确界，推导了|γ2|-|γ1|和|Γ2|-|Γ1|的精确上下界，并得到了与对数逆系数相关的第二Hankel行列式的精确估计。

Logarithmic coefficients of close-to-convex functions

For an analytic and univalent function $f$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$, the logarithmic coefficients $γ_n$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty} γ_n z^n$. In the present paper, we consider the class of close-to-convex functions (with argument $0$), and determine the sharp upper bound of $|γ_3|$ for such functions $f$, which proves a recent conjecture of the first and third authors [1].

Solution of the logarithmic coefficients conjecture in some families of univalent functions

For univalent and normalized functions $f$ the logarithmic coefficients $γ_n(f)$ are determined by the formula $\log(f(z)/z)=\sum_{n=1}^{\infty}2γ_n(f)z^n$. In the paper \cite{Pon} the authors posed the conjecture that a locally univalent function in the unit disk, satisfying the condition \[ \Re\left\{1+zf''(z)/f'(z)\right\}<1+λ/2\quad (z\in \mathbb{D}), \] fulfill also the following inequality: $$|γ_n(f)|\le λ/(2n(n+1)).$$ Here $λ$ is a real number such that $0<λ\le 1$. In the paper we confirm that the conjecture is true, and sharp.

Sharp inequalities for logarithmic coefficients and their applications

I. M. Milin proposed, in his 1971 paper, a system of inequalities for the logarithmic coefficients of normalized univalent functions on the unit disk of the complex plane. This is known as the Lebedev-Milin conjecture and implies the Robertson conjecture which in turn implies the Bieberbach conjecture. In 1984, Louis de Branges settled the long-standing Bieberbach conjecture by showing the Lebedev-Milin conjecture. Recently, O.~Roth proved an interesting sharp inequality for the logarithmic coefficients based on the proof by de Branges. In this paper, following Roth's ideas, we will show more …

苹果形凸子类 $\mathcal{C}_{\mathcal{AP}}$ 的逆对数系数及相关尖界估计

Inverse Logarithmic Coefficients and Associated Sharp Estimates for the Apple-Like Convex Subclass $\mathcal{C}_{\mathcal{AP}}$

研究苹果形凸子类C_AP的逆对数系数，得到初始逆对数系数Γ1、Γ2、Γ3及模差|Γ2|-|Γ1|的尖界，并给出二阶逆对数Hankel行列式、广义Fekete-Szegő泛函和三阶Hermitian-Toeplitz行列式的尖界。

与 Ma-Minda 类相关的反函数 Toeplitz 行列式及其对数系数

Toeplitz Determinants for Inverse Functions and their Logarithmic Coefficients Associated with Ma-Minda Classes

本文研究 Ma-Minda 型星形与凸函数类中反函数的系数及对数系数，给出相关 Toeplitz 行列式的精确界，并统一推广到多个已知子类。

与月牙形相关的星形函数和凸函数的对数系数的第二类Hankel行列式

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。

On logarithmic coefficients of some close-to-convex functions

The logarithmic coefficients $γ_n$ of an analytic and univalent function $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$ is defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty} γ_n z^n$. Recently, D.K. Thomas [On the logarithmic coefficients of close to convex functions, {\it Proc. Amer. Math. Soc.} {\bf 144} (2016), 1681--1687] proved that $|γ_3|\le \frac{7}{12}$ for functions in a subclass of close-to-convex functions (with argument $0$) and claimed that the estimate is sharp by providing a form of a extremal function. In the present paper,…

一类单叶函数对数系数的精确界

Sharp bounds of logarithmic coefficients for a class of univalent functions

针对满足给定解析不等式的单叶函数类 $\mathcal{U(α,λ)}$，本文研究其前三个对数系数并建立了精确上界。

星像函数与凸函数的指数类的对数系数

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。

关于Ma-Minda凸函数的初始对数系数与逆对数系数的模差

On the Moduli Difference of Initial Logarithmic and Inverse Logarithmic Coefficients of Ma-Minda Convex Functions

针对Ma-Minda凸函数子类，研究其初始对数系数与逆对数系数的模差，得到相关系数的精确估计，证明两类系数模差的不变性，统一了已有相关研究结果。

关于系数的对数下阶与 $\overline{\mathbb{C}}\setminus\{z_{0}\}$ 中复线性微分方程解的增长性之间的关系

On The Relationship Between The Logarithmic Lower Order of Coefficients and The Growth of Solutions of Complex Linear Differential Equations in $\overline{\mathbb{C}}\setminus\{z_{0}\}$

本文研究了 $\overline{\mathbb{C}}\setminus\{z_{0}\}$ 中齐次复线性微分方程解的增长性，在存在主导系数（由对数下阶或对数下型决定）的充分条件下，推广了 Liu、Long 和 Zeng 等人的先前结果。

Logarithmic Coefficients and a Coefficient Conjecture for Univalent Functions

Let ${\mathcal U}(λ)$ denote the family of analytic functions $f(z)$, $f(0)=0=f'(0)-1$, in the unit disk $\ID$, which satisfy the condition $\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<λ$ for some $0<λ\leq 1$. The logarithmic coefficients $γ_n$ of $f$ are defined by the formula $\log(f(z)/z)=2\sum_{n=1}^\infty γ_nz^n$. In a recent paper, the present authors proposed a conjecture that if $f\in {\mathcal U}(λ)$ for some $0<λ\leq 1$, then $|a_n|\leq \sum_{k=0}^{n-1}λ^k$ for $n\geq 2$ and provided a new proof for the case $n=2$. One of the aims of this article is to present a proof of this conjectur…

关于某些单叶函数类的对数系数第二Hankel行列式的精确界

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。

单值函数逆对数系数的模差

Moduli difference of inverse logarithmic coefficients of univalent functions

本文研究了单值函数类中逆对数系数的模差界，确定了第二和第一逆对数系数的精确上界和下界。

某类单叶函数的逆函数的对数系数的第二 Hankel 行列式

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。

Logarithmic singularity of the Szegö kernel and a global invariant of strictly pseudoconvex domains

The Szego kernel of a strictly pseudoconvex domain admits a singularity on the boundary diagonal, which consists of a pole and logarithmic type singularity. In this paper, we prove that the integral over the boundary of the coefficient of the logarithmic singularity gives a biholomorphic invariant of a domain, or a CR invariant of the boundary. We also show that the same invariant appears as the coefficient of the logarithmic term of the volume expansion of the domain with respect to the Bergman volume element.

有界旋转函数逆系数的第三Hankel行列式

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。

Ozaki 近凸函数系数泛函的精确估计

Sharp Estimates on Coefficient functionals of Ozaki close-to-convex functions

针对 Ozaki 近凸函数，本文研究并给出其对数系数相关二阶 Hermitian–Toeplitz 行列式、初始对数逆系数和初始阶 Schwarzian 导数等系数泛函的最佳可能估计。

Logarithmic coefficients of the inverse of univalent functions

Let $\es$ be the class of analytic and univalent functions in the unit disk $|z|<1$, that have a series of the form $f(z)=z+ \sum_{n=2}^{\infty}a_nz^n$. Let $F$ be the inverse of the function $f\in\es$ with the series expansion %in a disk of radius at least $1/4$ $F(w)=f^{-1}(w)=w+ \sum_{n=2}^{\infty}A_nw^n$ for $|w|<1/4$. The logarithmic inverse coefficients $Γ_n$ of $F$ are defined by the formula $\log\left(F(w)/w\right)\,=\,2\sum_{n=1}^{\infty}Γ_n(F)w^n$. % In this paper, we determine the logarithmic inverse coefficients bound of $F$ for the class In this paper, we first determine the sharp…

Entire Dirichlet series with monotonous coefficients and logarithmic h-measure

Let $F$ be an entire function represented by absolutely convergent for all $z\in\mathbb{C}$ Dirichlet series of the form $ F(z) = \sum\nolimits_{n=0}^{+\infty} a_{n}e^{zλ_{n}},$\ where a sequence $(λ_n)$ such that $λ_n\in\mathbb{R}\ \ (n\geq0)$, $λ_n\not=λ_k$ for any $n\not=k$ and $(\forall n\geq 0):\ 0\leqλ_n<β:=\sup\{λ_j:\ j\geq0\}\leq +\infty.$ {Let $h$ be non-decrease positive continuous function on $[0,+\infty)$ and $Φ$ increase positive continuous on $[0,+\infty)$ function.} In this paper we {find} the condition {on} $(μ_n)$ and $(λ_n)$ {such that} the relation $F(x+iy)=(1+o(1))a_{ν(x, F…

初等反函数的对数系数模差

Moduli difference of initial inverse logarithmic coefficients for starlike and convex functions

本文研究了星形函数和凸函数类中前两个反对数系数模差的精确上下界，利用子ordination技术与Schwarz函数系数估计，明确极值函数。

1/2阶星形函数逆系数第三Hankel行列式

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。

反向对数系数、差分、汉克尔行列式及费克特-塞戈函数在类$\mathcal{C}_e$中的应用

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。

某些单叶函数对数系数的尖锐不等式

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。

Logarithmic coefficients problems in families related to starlike and convex functions

Let $\es$ be the family of analytic and univalent functions $f$ in the unit disk $\D$ with the normalization $f(0)=f'(0)-1=0$, and let $γ_n(f)=γ_n$ denote the logarithmic coefficients of $f\in {\es}$. In this paper, we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families $\F(c)$ and $\G(δ)$ of functions $f\in {\es}$ defined by $$ {\rm Re} \left ( 1+\frac{zf''(z)}{f'(z)}\right )>1-\frac{c}{2}\, \mbox{ and } \, {\rm Re} \left ( 1+\frac{zf''(z)}{f'(z)}\right )<1+\fracδ{2},\quad z\in \D $$ for some $c\in(0,3]$ and $δ\in (0,1]$…

与归一化反正切映射相关的Ma-Minda凸类的系数问题

Coefficient Problems for a Ma-Minda Convex Class Associated with the Normalized Arcsine Mapping

本文针对归一化反正切映射生成的Ma-Minda凸类，利用从属技巧等开展系数分析，得到泰勒系数、Hankel行列式等的精确估计，确定对应极值函数，证明所得不等式为最优。

叶形星像函数类的系数估计与Hankel行列式

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。

Coefficient problems for certain Close-to-Convex Functions

In this paper, sharp bounds are established for the second Hankel determinant of logarithmic coefficients for normalised analytic functions satisfying certain differential inequality.

Grunsky系数在单叶函数理论中的某些应用

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。

关于星形函数逆系数的第三Hankel行列式：伯恩斯坦多项式方法

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。

关于 $S^*_ρ$ 的系数问题

On Coefficient problems for \textbf{$S^*_ρ$}

本文研究了星形函数类 $\mathcal{S}^*_ρ$ 的系数问题，基于函数及其逆函数的对数系数，建立了第二 Hankel 与 Toeplitz 行列式的界。

与气球形域相关的星形函数系数问题

Coefficient problems of Starlike Functions Related to a Balloon-Shaped Domain

本文研究了与气球形域相关的星形函数的系数问题，建立了二阶Hankel和Toeplitz行列式的边界。

关于Kaehler-Einstein流形epsilon函数的TYCZ展开中的第三系数

On the third coefficient in the TYCZ-expansion of the epsilon function of Kaehler-Einstein manifolds

该研究计算了Kaehler-Einstein度量相关epsilon函数的TYCZ展开第三系数，并探讨了该系数为零带来的相关结果，推进了相关几何展开的系数研究。

与Gregory系数相关的星形函数的系数界

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。

关于某类单叶函数的逆函数对数系数的 Hankel 和 Toeplitz 行列式

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。

关于\(\mathcal{S}^*_{q_1}\)类的系数问题与行列式的精确估计

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。

Asymptotics of Polynomials Orthogonal with respect to a Logarithmic Weight

In this paper we compute the asymptotic behavior of the recurrence coefficients for polynomials orthogonal with respect to a logarithmic weight $w(x){\rm d}x = \log \frac{2k}{1-x}{\rm d}x$ on $(-1,1)$, $k > 1$, and verify a conjecture of A.&nbsp;Magnus for these coefficients. We use Riemann-Hilbert/steepest-descent methods, but not in the standard way as there is no known parametrix for the Riemann-Hilbert problem in a neighborhood of the logarithmic singularity at $x=1$.