H. Mahzoon, R. Kargar
In the present paper we estimate the norm of the pre-Schwarzian derivative of bi-starlike functions of order $α$ where $α\in[0,1)$. Initially this problem was handled by Rahmatan et al. in [Bull Iran Math Soc {\bf43}: 1037-1043, 2017]. We pointed out that the proofs and bounds by Rahmatan et al. are incorrect and present correct proofs and bounds.
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In this paper, we consider the class of uniformly locally univalent harmonic mappings in the unit disk and build a relationship between its pre-Schwarzian norm and uniformly hyperbolic radius. Also, we establish eight ways of characterizing uniformly locally univalent sense-preserving harmonic mappings. We also present some sharp distortions and growth estimates and investigate their connections with Hardy spaces. Finally, we study subordination principles of norm estimates.
On a multiplier operator induced by the Schwarzian derivative of univalent functions
该研究考察由可拟共形延拓单叶函数的 Schwarz 导数诱导的乘子算子，并据此验证一大类拟圆盘满足 Brennan 猜想，同时给出渐近共形曲线和 Weil-Petersson 曲线的新刻画。
Let $Co(α)$ denote the class of concave univalent functions in the unit disk $\ID$. Each function $f\in Co(α)$ maps the unit disk $\ID$ onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional $(1-|z|^2)\left ( f''(z)/f'(z)\right)$, $f\in Co(α)$. In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional $(1-|z|^2)\left(f''(z)/f'(z)\right)$, $f\in Co(α)$ whenever $f''(0)$ is fixed. We also give a characterization for co…
We prove that if the Schwarzian norm of a given complex-valued locally univalent harmonic mapping $f$ in the unit disk is small enough, then $f$ is, indeed, globally univalent and can be extended to a quasiconformal mapping in the extended complex plane.
The primary aim of this article is to extend certain inequalities concerning the pre-Schwarzian derivatives from the case of analytic univalent functions to that of univalent harmonic mappings defined on certain domains. This is done in two different ways. One of the ways is to connect with a conjecture on the univalent harmonic mappings. Also, we improve certain known results on the majorization of the Jacobian of functions in the affine and linear invariant family of sense-preserving harmonic mappings. This is achieved as an application of a corresponding distortion theorem in terms of the h…
Quantitative estimates are obtained for the (finite) valence of functions analytic in the unit disk with Schwarzian derivative that is bounded or of slow growth. A harmonic mapping is shown to be uniformly locally univalent with respect to the hyperbolic metric if and only if it has finite Schwarzian norm, thus generalizing a result of B. Schwarz for analytic functions. A numerical bound is obtained for the Schwarzian norms of univalent harmonic mappings.
Sharp multiplier estimates for the higher-order Schwarzian derivatives of the Koebe function
本文研究加权Bergman空间上乘法算子的乘子范数估计，其符号为单叶函数的高阶Schwarz导数，建立了Koebe函数的高阶Schwarz导数的尖锐乘子估计，推广了Shimorin的相关结果。
We determine the representation theorem, distortion theorem, coefficients estimate and Bohr's radius for log-harmonic starlike mappings of order $α$, which are generalization of some earlier results. In addition, the inner mapping radius of log-harmonic mappings is also established by constructing a family of $1$-slit log-harmonic mappings. Finally, we introduce pre-Schwarzian, Schwarzian derivatives and Bloch's norm for non-vanishing log-harmonic mappings, several properties related to these are also obtained.
We review the relation between the classical formulas of the pre-Schwarzian and Schwarzian derivatives of locally univalent analytic functions and the derivatives of the generating functions of the methods due to Newton and Halley, respectively, for approximating zeros. We extend these relations to the cases when the functions considered are harmonic.
It is well-known that two locally univalent analytic functions $φ$ and $ψ$ have equal Schwarzian derivative if and only if there exists a non-constant Möbius transformation $T$ such that $φ=T\circ ψ$. In this paper, we identify completely the relationship between two locally univalent harmonic mappings with equal (harmonic) Schwarzian derivative.
In this article, we study preserving properties of certain nonlinear integral transforms in some classical families of normalized analytic univalent functions defined in the unit disk. Also, we find sharp pre-Schwarzian norm estimates of such integrals.
In this paper we introduce a definition of the pre-Schwarzian and the Schwarzian derivatives of any locally univalent harmonic mapping $f$ in the complex plane without assuming any additional condition on the (second complex) dilatation $ω_f$ of $f$. Using the new definition for the Schwarzian derivative of harmonic mappings, we prove analogous theorems to those by Chuaqui, Duren, and Osgood. Also, we obtain a Becker-type criterion for the univalence of harmonic mappings.
Some of important univalence criteria for a non-constant meromorphic function $f(z)$ on the unit disk $\ID$ involve its pre-Schwarzian or Schwarzian derivative. We consider an appropriate norm for the pre-Schwarzian derivative, and discuss the problem of finding the largest possible $r\in (0,1)$ for which the pre-Schwarzian norm of the dilation $r^{-1}f(rz)$ is not greater than a prescribed number for normalized univalent functions $f(z)$ in the unit disk. Similar results concerning the Schwarzian derivative are also obtained
On the pre-Schwarzian norm of certain Logharmonic mappings
本研究建立对数调和映射前施瓦茨范数与解析函数前施瓦茨范数的关联，推导局部单叶对数调和映射范数有限及对数调和函数为Bloch函数的充要条件，还得到对数调和Bloch映射及其解析、余解析部分的范数与增长定理。
In this paper, we extend Ahlfors's univalent criteria and Ahlfors's quasiconformal extension for analytic functions to harmonic mappings defined in the unit disk. Moreover, we give a general quasiconformal extension of harmonic Teichmüller mappings, whose maximal dilatation estimate is asymptotically sharp.
In this paper we study sharp estimates of pre-Schwarzian derivatives of functions belonging to the Nehari-type classes by using techniques from differential equations. In the sequel, we also see that a solution of a complex differential equation has a special form in terms of ratio of hypergeometric functions resulting to an integral representation. Finally, we attempt to study those univalent functions in the unit disk for which the image domain is an unbounded John domain.
On the pre-Schwarzian and Schwarzian derivatives of log-harmonic mappings
本文为局部单叶对数调和映射定义前施瓦茨导数与施瓦茨导数，研究其几何函数论性质，并给出有限前施瓦茨范数的充要条件及其与解析函数范数的关系。
We consider the family of all meromorphic functions $f$ of the form $$ f(z)=\frac{1}{z}+b_0+b_1z+b_2z^2+\cdots $$ analytic and locally univalent in the puncture disk $\mathbb{D}_0:=\{z\in\mathbb{C}:\,0<|z|<1\}$. Our first objective in this paper is to find a sufficient condition for $f$ to be meromorphically convex of order $α$, $0\le α<1$, in terms of the fact that the absolute value of the well-known Schwarzian derivative $S_f (z)$ of $f$ is bounded above by a smallest positive root of a non-linear equation. Secondly, we consider a family of functions $g$ of the form $g(z)=z+a_2z^2+a_3z^3+\c…
Pre-Schwarzian norm estimate and characterization of certain harmonic mappings
研究单位圆盘上某类调和映射，给出预Schwarzian范数估计，证明其单叶性和近凸性，并讨论相关解析与共轭解析部分的增长与偏差定理，最后给出解析部分的系数估计。
The notion of Schwarzian derivative for locally univalent holomorphic functions on complex plane was generalized for conformal diffeomorphisms by Osgood and Stowe in 1992 [27]. We shall identify a tensor that may serve as an analogue of the Schwarzian of Osgood and Stowe for CR mappings, and then use the tensor to define and study the CR Möbius transformations and metrics of pseudo-hermitian manifolds. We shall establish basic properties of the CR Schwarzian and a local characterization of CR spherical manifolds in terms of fully integrability of the CR Möbius equation. In another direction, w…
We study the order of affine and linear invariant families of planar harmonic mappings in the unit disk and determine the order of the family of mappings with bounded Schwarzian norm. The result shows that finding the order of the class $\mathcal{S}_H$ of univalent harmonic mappings can be formulated as a question about Schwarzian norm and, in particular, our result shows consistency between the conjectured order of $\mathcal{S}_H$ and the Schwarzian norm of the harmonic Koebe function.
A pre-Schwarzian and a Schwarzian derivative for locally univalent pluriharmonic mappings in ${\mathbb C}^n$ are introduced. Basic properties such as the chain rule, multiplicative invariance and affine invariance are proved for these operators. It is shown that the pre-Schwarzian is stable only with respect to rotations of the identity. A characterization is given for the case when the pre-Schwarzian derivative is holomorphic. Furthermore, it is shown that if the Schwarzian derivative of a pluriharmonic mapping vanishes then the analytic part of this mapping is a Möbius transformation. Some o…
Properties of Besov and $Q_p$ spaces in terms of the Schwarzian derivative of harmonic mappings
该研究借助Schwarzian导数与Carleson测度刻画单位圆盘内局部单叶保向调和映射的$\text{log }J_f$所属的Besov型与$Q_p$型空间，还引入基于雅可比算子的两类新空间并启动研究。
Higher-Order Schippers-Schwarzian Derivatives for Non-Circular Starlike Functions
该研究针对单位圆盘内由特定曲线界定区域从属关系定义的单叶星形函数子类，求得了高阶Schippers-Schwarzian导数的精确上界及对应极值函数，并将其应用于Grunsky系数精确界的推导。
Geometric subfamily of locally univalent functions, Blaschke products and quasidisk
本文研究了局部单值函数的几何子族F(α)，证明其与Blaschke乘积和Schwarzian导数的联系，并给出了准圆盘和预Schwarzian导数的精确界。
A normalized analytic function f is shown to be univalent in the open unit disk D if its second coefficient is sufficiently small and relates to its Schwarzian derivative through a certain inequality. New criteria for analytic functions to be in certain subclasses of functions are established in terms of the Schwarzian derivatives and the second coefficients. These include obtaining a sufficient condition for functions to be strongly $α$-Bazileviĉ of order $β$.
Pre-Schwarzian and Schwarzian norm estimates for harmonic functions with fixed analytic part
本文针对单位圆盘内局部单叶调和函数，证明了一般预Schwarzian范数估计并纠正旧错误，引入新类F0并给出其预Schwarzian和Schwarzian范数的最佳估计，以及余解析函数的畸变和系数估计。
We consider a univalent analytic function $f$ on the half-plane satisfying the condition that the supremum norm of its (pre-)Schwarzian derivative vanishes on the boundary. Under certain extra assumptions on $f$, we show that there exists a chordal Loewner chain initiated from $f$ until some finite time, and this Loewner chain defines a quasiconformal extension of $f$ over the boundary such that its complex dilatation is given explicitly in terms of the (pre-)Schwarzian derivative in some neighborhood of the boundary. This can be regarded as the half-plane version of the corresponding result d…
Pre-Schwarzian and Schwarzian norm Estimates for class of Ozaki Close-to-Convex functions
本文推导了Ozaki近凸函数的Schwarzian和pre-Schwarzian导数范数的精确界限，以及畸变和增长定理的精确界限，并探讨了解析部分固定的调和映射的pre-Schwarzian范数估计。
Generalized Schwarzians and Normal Families
本文研究有界广义Schwarzian导数的函数族，证明其拟正规性，推导新公式并应用于局部单叶函数族和整函数，经典情形为k=2。
Dirichlet symbols and the nonlinear wave equation
本文研究与单位圆盘压缩算子关联的狄利克雷符号，分析其满足的非线性波动方程，引入Schwarzian渐近方差并得到相关振幅的上界，扩展了通用Teichmüller空间的适用范围。
Pre-Schwarzian and Schwarzian norm Estimates for Robertson class
本文研究了Robertson类中满足特定条件的函数的预Schwarzian和Schwarzian范数估计，并给出了Schwarzian范数的精确上界。
Pre-Schwarzian and Schwarzian norm estimates for certain classes of analytic and harmonic mappings
本文研究了特定解析和调和映射类的Schwarzian导数和范数，建立了Schwarzian范数的精确估计，并通过预Schwarzian和Schwarzian范数得出单值性准则。
We establish a sharp norm estimate of the Schwarzian derivative for a function in the classes of convex functions introduced by Ma and Minda [Proceedings of the Conference on Complex Analysis, International Press Inc., 1992, 157-169]. As applications, we give sharp norm estimates for strongly convex functions of order $α,~0<α<1,$ and for uniformly convex functions.
For $-1\leq B<A\leq 1$, let $\mathcal{S}^*(A,B)$ denote the class of Janowski starlike functions which satisfy the subordination relation $zf'(z)/f(z)\prec (1+Az)/(1+Bz)$. In the present article, we determine the sharp of pre-Schwarzian norm for the functions in the class $\mathcal{S}^*(A,B)$.
In this paper, we obtain sharp bounds for the norm of pre--Schwarzian derivatives of certain analytic functions. Initially this problem was handled by H. Rahmatan, Sh. Najafzadeh and A. Ebadian [Stud Univ Babeş--Bolyai Math {\bf61}(2): 155--162, 2016]. We pointed out that the proofs by Rahmatan et al. are incorrect and present correct proofs.
Pre-Schwarzian norm estimate for certain Ma-Minda Class of functions
本文研究满足特定从属关系的 Ma-Minda 解析函数类，在两种特殊参数函数下给出函数及其 Alexander 变换的前 Schwarz 导数范数精确估计。
Growth, Distortion, and Schwarzian Norm Estimates for Exponentially Convex Functions
本文研究指数凸函数类的增长、畸变及Schwarzian范数估计，通过显式Schwarz函数推导参数依赖界，并分析极值问题。
Growth, Distortion, Pre-Schwarzian and Schwarzian norm estimates for Generalized Robertson class
本文研究广义罗伯特类函数的几何性质，推导其施瓦茨ian和预施瓦茨ian导数范数的精确界，并探讨函数的增长、扭曲及凹凸半径问题。
For $0<λ\le 1$, let $\mathcal{U}(λ)$ be the class analytic functions $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}$ satisfying $|f'(z)(z/f(z))^2-1|<λ$ and $\mathcal{U}:=\mathcal{U}(1)$. In the present article, we prove that the class $\mathcal{U}$ is contained in the closed convex hull of the class of starlike functions and using this fact, we solve some extremal problems such as integral mean problem and arc length problem for functions in $\mathcal{U}$. By means of the so-called theory of star functions, we also solve the integral mean problem for functions in $\mathcal{U…
The Grunsky norm of univalent functions and abelian holomorphic differentials
本研究证明圆盘上归一化单叶函数的Grunsky范数由全纯阿贝尔微分的平方决定，区别于关联所有可积全纯二次微分的Teichmuller范数，该结果可给出所有拟共形曲线Fredholm特征值的显式表示。
Sharp Pre-Schwarzian Norm Bounds for Ma-Minda Starlike Classes
本文建立统一框架计算 Ma-Minda 星形类的预 Schwarz 范数，给出直接通用方法，推导出经典和新引入星形子类的显式范数公式。
On the pre-Schwarzian norm estimate of special close-to-convex harmonic mappings
研究单位圆盘内具有特殊解析部分的近凸调和映射类，得到了精确的预Schwarzian范数估计，并讨论了调和Bloch映射理论及解析与共解析部分的增长与偏差定理。
On the Pre-Schwarzian Norm and Starlikeness of Certain Logharmonic Mappings
本文研究单位圆盘内的一类对数调和映射，得到其Pre-Schwarzian范数的精确界，讨论增长定理与星形性，推导遗传星形性的充分系数条件，并给出相关实例。
Let $\mathcal{A}$ denote the class of analytic functions $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ normalized by $f(0)=0$, $f'(0)=1$. In the present article, we obtain the sharp estimates of the Schwarzian norm for functions in the classes $\mathcal{G}(β)=\{f\in \mathcal{A}:{\rm Re\,}[1+zf''(z)/f'(z)]<1+β/2\}$, where $β>0$ and $\mathcal{F}(α)=\{f\in \mathcal{A}:{\rm Re\,}[1+zf''(z)/f'(z)]>α\}$, where $-1/2\le α\le 0$. We also establish two-point distortion theorem for functions in the classes $\mathcal{G}(β)$ and $\mathcal{F}(α)$.
We discuss the problem of embedding univalent functions into Loewner chains in higher dimension. In particular, we prove that a normalized univalent map of the ball in $\C^n$ whose image is a smooth strongly pseudoconvex domain is embeddable into a normalized Loewner chain (satisfying also some extra regularity properties) if and only if the closure of the image is polynomially convex.
Schwarzian Norm Estimate for Functions in Robertson Class
本文研究 Robertson 类 $\mathcal{S}_α$ 中规范化解析函数，给出并确定其 pre-Schwarzian 与 Schwarzian 范数的 sharp 估计。
For a real constant $b,$ we give sharp estimates of $\log|f(z)/z|+b\arg[f(z)/z]$ for subclasses of normalized univalent functions $f$ on the unit disk.
Schwarzian norm estimates for some classes of analytic and harmonic mappings
本文研究了单位圆盘内满足特定实部条件的规范化解析函数类$\u27a9(β)$，建立了以$f''(0)$为参数的前Schwarzian范数和Schwarzian范数的精确界，得到了该类函数的增长、偏差及半径结果，并将其应用于调和映射的前Schwarzian范数估计。
An estimation of the pre-Schwarzian norm for certain classes of analytic functions
本文研究了当φ(z)=1/(1-z)^s(0<s≤1)和φ(z)=(1+sz)^2(0<s≤1/√2)时，函数类S*(φ)和C(φ)的预Schwarz范数的精确估计。
Geometric subfamily of functions convex in some direction and Blaschke products
本文研究满足特定规范化与曲率条件的局部单叶解析函数族，证明其具有涉及 Blaschke 乘积、Schwarz 导数及单叶调和映射的若干优雅性质。
Sharp Bounds for Higher-Order Schippers Functionals Associated with Lune and Bean Domains
该研究针对与月牙域、豆形域相关的单叶函数子类，推导了三阶、四阶Schippers泛函的精确界，确定了极值函数与像域几何，还得到了初始Grunsky系数的精确估计。
The Schwarzian norm estimates for Janowski convex functions
本文研究Janowski凸函数类$\mathcal{C}(A,B)$的Schwarzian范数估计问题，借助Dieudonné引理得到精确估计结果，并通过新方法构造出达到该精确界的极值函数。
We consider support points of the class $S^0(\mathbb{D}^n)$ of normalized univalent mappings on the polydisc $\mathbb{D}^n$ with parametric representation and we prove sharp estimates for coefficients of degree 2.
Pre-Schwarzian norm estimation for functions in the Ma-Minda-type starlike and convex classes
本文针对三类参数条件下的Ma-Minda型星形类与凸类函数，建立了预Schwarzian范数的精确估计。
Schwarzian Norm Estimate for Functions in Generalized Robertson Class
本文研究广义 Robertson 类 $\mathcal{S}_α(β)$，针对满足给定旋转实部不等式的规范化解析函数，推导其 pre-Schwarzian 与 Schwarzian 范数的 sharp 估计。
In this paper, we introduce the subclass $SHP^{-m}(α,β)$ using integral operator and give sufficient coefficient conditions for normalized harmonic univalent function in the subclass $SHP^{-m}(α,β)$.These conditions are also shown to be necessary when the coefficients are negative.
Estimates are obtained for the initial coefficients of a normalized analytic function $f$ in the unit disk $\mathbb{D}$ such that $f$ and the analytic extension of $f^{-1}$ to $\mathbb{D}$ belong to certain subclasses of univalent functions. The bounds obtained improve some existing known bounds.
The main purpose of this paper is to obtain sharp bounds of the norm of Schwarzian derivative for convex mappings of order $alpha$ in terms of the value of $f''(0)$, in particular, when this quantity is equal to zero. In addition, we obtain sharp bounds for distortion and growth for this mappings and we generalized the results obtained by Suita and Yamashita for this particular case.
For a normalised analytic function f defined on the open unit disk in the complex plane, we determine several sufficient conditions for starlikeness in terms of the quotients Q_{ST}:=zf'(z)/f(z), Q_{CV}:=1+zf"(z)/f'(z) and the Schwarzian derivative Q_{SD}:=z^2((f"(z)/f'(z))'-(f"(z)/f'(z))^2/2)$. These conditions were obtained by using the admissibility criteria of starlikeness in the theory of second order differential subordination.
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